Methods and apparatus for estimating total unique audiences

ABSTRACT

Methods and apparatus for determining a unique audience exposed to media while reducing memory resources of a computing device are disclosed herein. Example instructions cause a machine to at least, based on impression requests from a plurality of client devices via a network, log a plurality of impressions corresponding to media accessed at the client devices; obtain a count of demographic impressions logged by a database proprietor; obtain a count of registered users of the database proprietor exposed to the media; and execute a process to determine a unique audience size by multiplying a count of the plurality of impressions by a square of the count of the registered users to generate a product; dividing the product by the count of the demographic impressions to generate a quotient; and determining the unique audience size based on a square root of the quotient.

CROSS REFERENCE TO RELATED APPLICATIONS

This Patent arises from a continuation of U.S. patent application Ser.No. 16/388,666, filed on Apr. 18, 2019, entitled “METHODS AND APPARATUSFOR ESTIMATING TOTAL UNIQUE AUDIENCES,” which is a continuation of U.S.patent application Ser. No. 15/008,220, filed on Jan. 27, 2016, now U.S.Pat. No. ______, and entitled “METHODS AND APPARATUS FOR INCREASING THEROBUSTNESS OF MEDIA SIGNATURES.” Priority to U.S. patent applicationSer. No. 16/388,666 and U.S. patent application Ser. No. 15/008,220 isclaimed. U.S. patent application Ser. No. 16/388,666 and U.S. patentapplication Ser. No. 15/008,220 are hereby incorporated herein byreference in their entireties.

FIELD OF THE DISCLOSURE

This disclosure relates generally to monitoring media and, moreparticularly, to methods and apparatus for estimating total uniqueaudiences exposed to media.

BACKGROUND

Traditionally, audience measurement entities have measured audienceengagement levels for media based on registered panel members. That is,an audience measurement entity (AME) enrolls people who consent to beingmonitored into a panel. The AME then monitors those panel members todetermine media (e.g., television programs, radio programs, movies,DVDs, advertisements, streaming media, websites, etc.) presented tothose panel members. In this manner, the AME can determine exposuremetrics for different media based on the collected media measurementdata.

Techniques for monitoring user access to Internet resources, such aswebpages, advertisements and/or other Internet-accessible media, haveevolved significantly over the years. Internet-accessible media is alsoknown as online media. Some known systems perform such monitoringprimarily through server logs. In particular, entities serving media onthe Internet can use known techniques to log the number of requestsreceived at their servers for media.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1C illustrate example computer resource consumption for threeexample processes for determining a unique audience for particular mediabased on impressions and/or a determining frequency distribution of theimpressions based on example data inputs.

FIG. 2 illustrates example client devices that report audienceimpression requests for Internet-based media to impression collectionentities to facilitate identifying total impression requests and sizesof audiences exposed to different Internet-based media.

FIG. 3 is a block diagram of the example audience/impression determinerof FIG. 2.

FIGS. 4-7 are flowcharts representative of example machine readableinstructions that may be executed to implement the audience/impressiondeterminer of FIG. 2 and/or FIG. 3 to determine the unique audience forparticular media based on impressions and/or the frequency distributionof the impressions.

FIGS. 8A-8D illustrate example data associated with a second process ofthe example processes of FIGS. 1A-1C used by the example audience datacalculator and/or the example impression data calculator of FIG. 3 todetermine the unique audience and the frequency distribution of theimpressions.

FIG. 9 illustrates example data associated with a third process of theexample processes of FIGS. 1A-1C used by the example audience datacalculator and/or the example impression data calculator of FIG. 3 todetermine the unique audience and the frequency distribution ofimpression requests.

FIG. 10 is a block diagram of an example processor platform that may beutilized to execute the example instructions of FIGS. 4-7 to implementthe example audience/impression determiner of FIG. 2 and/or FIG. 3.

The figures are not to scale. Wherever possible, the same referencenumbers will be used throughout the drawing(s) and accompanying writtendescription to refer to the same or like parts

DETAILED DESCRIPTION

Techniques for monitoring user access to Internet-accessible media, suchas websites, advertisements, content and/or other media, have evolvedsignificantly over the years. Internet-accessible media is also known asonline media. In the past, such monitoring was done primarily throughserver logs. In particular, entities serving media on the Internet wouldlog the number of requests received for their media at their servers.Basing Internet usage research on server logs is problematic for severalreasons. For example, server logs can be tampered with either directlyor via zombie programs, which repeatedly request media from the serverto increase the server log counts. Also, media is sometimes retrievedonce, cached locally and then repeatedly accessed from the local cachewithout involving the server. Server logs cannot track such repeat viewsof cached media. Thus, server logs are susceptible to both over-countingand under-counting errors.

The inventions disclosed in Blumenau, U.S. Pat. No. 6,108,637, which ishereby incorporated herein by reference in its entirety, fundamentallychanged the way Internet monitoring is performed and overcame thelimitations of the server side log monitoring techniques describedabove. For example, Blumenau disclosed a technique wherein Internetmedia to be tracked is tagged with monitoring instructions. Inparticular, monitoring instructions (also known as a media impressionrequest) are associated with the hypertext markup language (HTML) of themedia to be tracked. When a client requests the media, both the mediaand the impression request are downloaded to the client. The impressionrequests are, thus, executed whenever the media is accessed, be it froma server or from a cache.

Impression requests cause monitoring data reflecting information aboutan access to the media to be sent from the client that downloaded themedia to a monitoring entity. Sending the monitoring data from theclient to the monitoring entity is known as an impression request.Typically, the monitoring entity is an AME that did not provide themedia to the client and who is a trusted (e.g., neutral) third party forproviding accurate usage statistics (e.g., The Nielsen Company, LLC).Advantageously, because the impression requests are associated with themedia and executed by the client browser whenever the media is accessed,the monitoring information is provided to the AME (e.g., via animpression request) irrespective of whether the client corresponds to apanelist of the AME.

There are many database proprietors operating on the Internet. Thesedatabase proprietors provide services to large numbers of subscribers.In exchange for the provision of services, the subscribers register withthe database proprietors. Examples of such database proprietors includesocial network sites (e.g., Facebook, Twitter, MySpace, etc.),multi-service sites (e.g., Yahoo!, Google, Axiom, Catalina, etc.),online retailer sites Amazon.com, Buy.com, etc.), credit reporting sites(e.g., Experian), streaming media sites (e.g., YouTube, etc.), etc.These database proprietors set cookies and/or other device/useridentifiers on the client devices of their subscribers to enable thedatabase proprietor to recognize their subscribers when they visit theirwebsite.

The protocols of the Internet make cookies inaccessible outside of thedomain (e.g., Internet domain, domain name, etc.) on which they wereset. Thus, a cookie set in, for example, the amazon.com domain isaccessible to servers in the amazon.com domain, but not to serversoutside that domain. Therefore, although an AME might find itadvantageous to access the cookies set by the database proprietors, theyare unable to do so.

The inventions disclosed in Mainak et al., U.S. Pat. No. 8,370,489,which is incorporated by reference herein in its entirety, enable an AMEto leverage the existing databases of database proprietors to collectmore extensive Internet usage by extending the impression requestprocess to encompass partnered database proprietors and by using suchpartners as interim data collectors. The inventions disclosed in Mainaket al. accomplish this task by structuring the AME to respond toimpression requests from clients (who may not be a member of an audiencemember panel and, thus, may be unknown to the audience member entity) byredirecting the clients from the AME to a database proprietor, such as asocial network site partnered with the audience member entity, using animpression response. Such a redirection initiates a communicationsession between the client accessing the tagged media and the databaseproprietor. For example, the impression response received from the AMEmay cause the client to send a second impression request to the databaseproprietor. In response to receiving this impression request, thedatabase proprietor (e.g., Facebook) can access any cookie it has set onthe client to thereby identify the client based on the internal recordsof the database proprietor. In the event the client corresponds to asubscriber of the database proprietor, the database proprietorlogs/records a database proprietor demographic impression in associationwith the client/user and subsequently forwards logged databaseproprietor demographic impressions to the AME.

As used herein, an impression is defined to be an event in which a homeor individual accesses and/or is exposed to media (e.g., anadvertisement, content, a group of advertisements and/or a collection ofcontent). In Internet advertising, a quantity of impressions orimpression count is the total number of times media (e.g., content, anadvertisement or advertisement campaign) has been accessed by a webpopulation (e.g., the number of times the media is accessed). In someexamples, an impression or media impression is logged by an impressioncollection entity (e.g., an AME or a database proprietor) in response toa beacon request from a user/client device that requested the media. Insome examples, a media impression is not associated with demographics. Apanelist demographic impression is a media impression logged by an AMEfor which the AME has panelist demographics corresponding to a householdand/or audience member exposed to media. As used herein, a databaseproprietor demographic impression is an impression recorded by adatabase proprietor in association with corresponding demographicinformation provided by the database proprietor in response to a beaconrequest from a client device of a registered subscriber of the databaseproprietor.

In the event the client does not correspond to a subscriber of thedatabase proprietor, the database proprietor may redirect the client tothe AME and/or another database proprietor. If the client is redirectedto the AME, the AME may respond to the redirection from the firstdatabase proprietor by redirecting the client to a second, differentdatabase proprietor that is partnered with the AME. That second databaseproprietor may then attempt to identify the client as explained above.This process of redirecting the client from database proprietor todatabase proprietor can be performed any number of times until theclient is identified and the media exposure logged, or until alldatabase partners have been contacted without a successfulidentification of the client. In some examples, the redirections occurautomatically so the user of the client is not involved in the variouscommunication sessions and may not even know they are occurring.

Periodically or aperiodically, the partnered database proprietorsprovide their logs and demographic information to the AME, which thencompiles the collected data into statistical reports identifyingaudience members for the media.

Example techniques disclosed herein use database proprietors to identifyaudience demographics based on impression requests from client devicesto track quantities of impressions attributable to users of those clientdevices. In some examples, the database proprietor demographicimpressions collected by a database proprietor (e.g., Facebook, Yahoo,Google, etc.) may be inaccurate and/or incomplete when the databaseproprietor does not have complete coverage of device/user identifierscookies) at all of the client devices associated with impressionrequests or, more generally associated with an impression to be logged.As used herein in this context, coverage represents the extent to whicha database proprietor has set cookies or, more generally, device/useridentifiers in client devices associated with beacon requests. Forexample, if only 50% of client devices that send an impression requestassociated with a media impression to the database proprietor have acookie set therein by the database proprietor, then the databaseproprietor has 50% coverage of such client devices. A client device maynot have a cookie set by the database proprietor in its web browser if,for example, a user does not have an account with the databaseproprietor or if the user has an account with the database proprietorbut has cleared the cookie cache and deleted the database proprietor'scookie before or at the time of a media exposure. In yet other examples,the database proprietor may set a cookie on the client device but theclient device does not correspond to a registered user of the databaseproprietor. In any of such examples, the database proprietor would notbe able to identify the user associated with one or more mediaimpressions and, thus, would not report any database proprietordemographic impressions for those impressions.

Examples to estimate a unique audience size for logged media impressionsbased on logged database proprietor demographic impressions aredisclosed herein. In some examples, estimates of the unique audiencesize are determined from database proprietor demographic impression datacollected by database proprietors. In some disclosed examples, an AMEestimates a unique audience size using a number of media impressions, anumber of recorded (e.g., logged) database proprietor demographicimpressions, a frequency distribution of the recorded databaseproprietor demographic impressions across a partial audience and thenumber of people associated with the database proprietor demographicimpressions (e.g., the partial audience). The number of recordeddatabase proprietor demographic impressions and the partial audiencesize can be determined from the frequency distribution of the recordeddatabase proprietor demographic impressions. As used herein, a frequencydistribution is indicative of (1) a total quantity of unique audiencemembers who have not been exposed to a particular media, (2) a totalquantity of unique audience members who have been exposed to theparticular media exactly once, (3) a total quantity of unique audiencemembers who have been exposed to the particular media exactly twice,etc.

The people associated with the database proprietor demographicimpressions at the database proprietor are referred to as the partialaudience. The term partial audience is used because some individualsassociated with the media impression requests sent to the AME, may notbe registered with the database proprietor. As such, the databaseproprietor will not record (e.g., log) media impressions for theseindividuals in response to impression requests redirected by the AME tothe database proprietor because these individuals are not registeredwith the database proprietor.

In some disclosed examples, an AME sends a list of logged impressionsfor particular online media to one or more database proprietor(s). Thedatabase proprietor(s) respond with a number of recorded databaseproprietor demographic impressions from the partial audience, and thesize of the partial audience. In other examples, the database proprietormay receive media impression requests for media directly from clientdevices (e.g., without being redirected by the AME) that access themedia via one or more websites. In some examples, accessing media mayinclude media retrieved from a server through a website in response to auser-request specifically requesting the media. In some examples, themedia could be delivered by a server for presentation via a websitewithout a user intentionally requesting the media. For example, somemedia is presented on a website as a result of the website beingprogrammed to request and present the media as part of the website beingrendered. The database proprietor may record a quantity of mediaimpressions (e.g., impressions that are not matched with a user of thedatabase proprietor) and a quantity of database proprietor demographicimpressions (e.g., impressions that are matched with a user of thedatabase proprietor). In these other examples, the database proprietorwill provide the total quantity of media impressions not matched to auser of the database proprietor and the total quantity of databaseproprietor demographic impressions (e.g. the partial audience) to theAME.

Using examples disclosed herein, the AME determines an estimate size ofan audience based on logged impressions using techniques designed tooptimize computer resources (e.g., processor resources and memoryresources) based on the number and/or complexity (e.g., number ofwebsites associated with the logged impressions and/or number of loggedimpressions per user) of the logged impressions. In particular, threedifferent UA/FD processes are disclosed herein to estimate uniqueaudience sizes. The example UA/FD processes have different trade-offsbetween memory resource usage and processor resource usage underdifferent circumstances. For example, a first unique audience and/orfrequency distribution process, herein referred to as “UA/FD process 1,”requires the least amount of processor resources (e.g., is leastcomputationally intensive) and requires the least amount of memoryresources. Example UA/FD process 1 is configured to estimate uniqueaudiences based on logged media impressions and database proprietordemographic impressions associated with media accessed via a website. Asecond unique audience and/or frequency distribution process, hereinreferred to as “UA/FD process 2,” requires more processor resources andmemory resources than UA/FD process 1. Example UA/FD process 2 isconfigured to estimate unique audiences based on logged impressions anddatabase proprietor demographic impressions associated with mediaaccessed via one or more websites. Although UA/FD process 2 requiresmore processor resources and memory resources, the estimates from UA/FDprocess 2 are more accurate than UA/FD process 1 because UA/FD process 2uses data (e.g., logged impressions) corresponding to media accesses viamore than one website to estimate unique audiences. A third uniqueaudience and/or frequency distribution process, herein referred to as“UA/FD process 3,” requires the most processor resources, but lessmemory resources than UA/FD process 2. UA/FD process 3 is designed toestimate unique audiences based on logged impressions and audience datacorresponding to media accesses via one or more websites.

In some examples, although UA/FD process 3 requires the most processorresources, UA/FD process 3 is usable under certain situations in whichUA/FD processes 1 and 2 disclosed herein are not capable to determineunique audience. For example, the AME may receive impression requestsand database proprietor demographic impressions associated with mediaaccessed via hundreds of websites, where each person may have beenexposed to the media hundreds of times. In such an example, UA/FDprocess 1 and UA/FD process 2 may not be usable to calculate the uniqueaudience due to the large number of websites and/or impressions. Forexample, UA/FD process 1 is only used to determine a unique audiencebased on impressions corresponding to media accessed via one website. Inaddition, UA/FD process 2 may not be usable because a processor system(e.g., a computer) may not have sufficient available memory resourcesfor UA/FD process 2 to process the large number of impressions from thehundreds of websites through which the media was accessed. In such anexample, by adjusting for available processor resources and memoryresources, UA/FD process 3 may be the relatively best solution of thethree UA/FD processes disclosed herein to estimate a unique audience forthe media accessed via the hundreds of websites.

FIGS. 1A-1C include example situations 110, 115, 120, 125, 130, 135,140, 145, 150 including example media 100, an example UA/FD process 1101, an example UA/FD process 2 102, an example UA/FD process 3 103,example processor resources 104, example memory resources 105, anexample webpage 106 example webpages 108, and example webpages 109.FIGS. 1A-1C illustrate amounts of the example processor resources 104and the example memory resources 105 required for different ones of theexample situations 110-150 based on the example UA/FD processes 101,102, 103. The list of example situations is not an exhaustive list ofpossible situations that can be handled by the three example UA/FDprocesses.

As described above, the example UA/FD process 1 101 requires the leastamount of the example processor resources 104 and requires the leastamount of the example memory resources 105. The example UA/FD process 1101 uses the principle of maximum entropy and minimum cross entropy.Given (1) an unspecified univariate distribution (A) with unknownprobabilities, q_(k), (where k can be any on-negative integer), (2) aknown expected value E[A]=μ₁, and (3) a known initial probabilityq_(o)=P[A=0], the principle of maximum entropy is used to determine theunspecified univariate distribution (A). To determine the unspecifiedunivariate distribution (A), Equation 1 below is determined.

maximize Q, H=−Σ _(k=0) ^(∞) log(q _(k)),   Equation 1

subject to q₀ given Σ_(k=0) ^(∞)q_(k)=1 and Σ_(k=0) ^(∞)kq_(k)=μ₁

The solution is a zero-modified geometric distribution of the form q₀given q_(k)=Cr^(k) where k=1, 2, . . . , ∞,

${C = \frac{\left( {1 - q_{0}} \right)^{2}}{\mu_{1} + q_{0} - 1}},{{{and}\mspace{14mu} r} = {\frac{\mu_{1} + q_{0} - 1}{\mu_{1}}.}}$

Once the unspecified univariate distribution (A) is determined, theprinciple of minimum cross entropy is used to determine a secondunspecified univariate distribution (B) with probabilities p_(k) on thesame mathematical domain as the first unspecified univariatedistribution (A), with constraint E[B]=μ₂. Calculating for the initialprobability p₀=P[B=0] results in a simple and accurate estimation for aunique audience based on logged impressions corresponding to mediaassociated with one website.

The solution becomes a previous distribution to a minimize cross entropyproblem (e.g., q is the prior distribution and p is the unknowndistribution to be solved) as shown in Equation 2 below.

$\begin{matrix}{{{minimize}\mspace{14mu} P},{{D\left( {P\text{:}Q} \right)} = {{p_{0}\mspace{14mu} {\log \left( \frac{p_{0}}{q_{0}} \right)}} + {\sum\limits_{k = 1}^{\infty}\; {p_{k}{\log \left( \frac{p_{k}}{{Cr}^{k}} \right)}}}}},\mspace{76mu} {{{subject}\mspace{14mu} {to}\mspace{14mu} {\sum\limits_{k = 0}^{\infty}\; p_{k}}} = {{1\mspace{14mu} {and}\mspace{14mu} {\sum\limits_{k = 0}^{\infty}\; {kp}_{k}}} = \mu_{2}}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

The solution is a zero-modified geometric distribution:

p₀=s₀q₀, p_(k)=s₀s₁ ^(k)Cr^(k), where k=1, 2, . . . , ∞.

In the illustrated example, s₀ and s₁ are solved to satisfy theconstraint Σ_(k=0) ^(∞)p_(k)=1 and Σ_(k=0) ^(∞)kp_(k)=μ₂). Thezero-modified geometric distribution (p_(k)) is a unique solution. Thedistribution (p_(k)) is plugged into the first constraint (e.g., Σ_(k=0)^(∞)p_(k)=1) and solved for s to determine Equation 3 below.

$\begin{matrix}{{s_{1} = \frac{{p_{0}q_{0}} - q_{0}}{r\left( {{- {Cp}_{0}} + {p_{0}q_{0}} - q_{0}} \right)}},} & {{Equation}\mspace{14mu} 3}\end{matrix}$

Substituting s₁ back into p_(k) and applying the second constraint(e.g., Σ_(k=0) ^(∞)kp_(k)=μ₂) results in Equation 4 below μ₂:

$\begin{matrix}{{\mu_{2} = \frac{\left( {p_{0} - 1} \right)\left( {{p_{0}\left( {{\mu_{1}q_{0}} + q_{0} - 1} \right)} - {q_{0}\left( {\mu_{1} + q_{1} - 1} \right)}} \right)}{{p_{0}\left( {q_{0} - 1} \right)}^{2}}},} & {{Equation}\mspace{14mu} 4}\end{matrix}$

Rearranging (e.g., using known mathematical properties) Equation 4results in Equation 5 below:

$\begin{matrix}{{\frac{\left( {\frac{\mu_{1}}{1 - q_{0}} - 1} \right)}{\left( \frac{1 - q_{0}}{q_{0}} \right)} = \frac{\left( {\frac{\mu_{2}}{1 - p_{0}} - 1} \right)}{\left( \frac{1 - p_{0}}{p_{0}} \right)}},} & {{Equation}\mspace{14mu} 5}\end{matrix}$

The example UA/FD process 1 101 estimates a unique audience (X) in auniverse population (U) that was exposed to the example media 100 (e.g.,by sending an impression request associated with the media) usingEquation 5 above. Additionally, a total number of logged mediaimpressions (T), a total number of database proprietor demographicimpressions (e.g., logged media impressions matched to a user of adatabase proprietor) (R), and a total identified database proprietoraudience exposed to the media 100 (A) are also utilized in Equation 5above. The example UA/FD process 1 101 equates

${{q_{0}\mspace{14mu} {to}\mspace{14mu} 1} - \frac{A}{U}},{\mu_{1}\mspace{14mu} {to}\mspace{14mu} \frac{R}{U}},{{p_{0}\mspace{14mu} {to}\mspace{14mu} 1} - \frac{X}{U}},{{and}\mspace{14mu} \mu_{2}\mspace{14mu} {to}\mspace{14mu} \frac{T}{U}}$

to produce a first unique audience sub-process

$\frac{\left( {R - A} \right)\left( {U - A} \right)}{A^{2}} = \frac{\left( {T - X} \right)\left( {U - X} \right)}{X^{2}}$

which is used by UA/FD process 1 101 to determine the total uniqueaudience (X).

In some examples, when the universe is large (e.g., larger than athreshold size) and the number of logged impressions (e.g., R and T) issmall (e.g., based on a threshold) relative to the audience sizes (e.g.,A and X), the first unique audience sub-process can be simplified to asecond unique audience sub-process. The second unique audiencesub-process is

$\frac{A^{2}}{R - A} = {\frac{X^{2}}{T - X}.}$

In some examples, when the universe is large and the number of loggedimpressions (e.g., R and T) are large (e.g., based on the threshold)relative to the audience sizes (e.g., A and X), the second uniqueaudience sub-process can be simplified to a third unique audiencesub-process. The third unique audience sub-process is

$\frac{A^{2}}{R} = {\frac{X^{2}}{T}.}$

The example UA/FD process 1 101 applies data associated with theimpressions of a particular media to one of the first, second, or thirdunique audience sub-processes based on the thresholds. Additionally, theexample UA/FD process 1 101 can determine the number of people in theunique audience associated with exactly one logged impression, exactlytwo logged impressions, etc. (e.g., a frequency distribution) based on ageometric distribution formula (e.g., P(Z=k)=(1−p)^(k−1)p, where

$p = \frac{X}{T}$

and k∈{1, 2, . . . , ∞}).

Since the three equations are simple to compute, processor resourcesrequired for the example UA/FD process 1 101 to determine a uniqueaudience are low. Additionally, since only A, R, T, and U are requiredto be stored in memory, required memory for the example UA/FD process 1101 is low.

As described above, the example UA/FD process 2 102 requires moreprocessor resources and memory than the example UA/FD process 1 101.However, the example UA/FD process 2 102 can determine a unique audiencebased on data from one or more websites associated with one or moredatabase proprietors. Additionally, if the estimate is based on two ormore websites, the estimate is more accurate. The example UA/FD process2 102 is derived from the process of maximum entropy and minimum crossentropy. The example UA/FD process 2 102 constructs anaudience/impressions model constraint matrix (e.g., C_(Q)) and anaudience/impressions total constraint vector (e.g., D_(Q)) to representthe total audience exposed to media based on one or more loggedimpressions. The example audience/impressions representation constraint(C_(Q)) matrix includes rows on constraints associated with a universepopulation, a total audience, an expected value of the total mediaimpressions, etc., as further described below in connection with FIG. 6.

The unique audiences exposed to media associated with logged impressionscan be determined by solving for an impression characteristics columnvector (Q). The impressions characteristics column vector (Q) includesprobabilities representing a number or people associated with zeroimpressions corresponding to website A and/or website B, one impressionfrom website A and/or website B, two impressions from website A and/orwebsite B, and/or any combination thereof. The example UA/FD process 2102 determines impressions characteristic column vector (Q) usingEquation 6 below.

maximize Q, H=−Σ _(k=0) ^(∞) q _(k) log(q _(k)),   Equation 6

subject to C_(Q)Q=D_(Q).

To solve the total unique audience for logged impressions associatedwith the example media 100, the example UA/FD process 2 102 usesimpressions characteristics column vector (Q) as the previousdistribution for estimating the same distribution of probabilities basedon audience (e.g., population) characteristics (e.g., P) using differentconstraints. The audience characteristics (P) are used to determine thetotal unique audience, as further described in FIG. 6. To determine theaudience characteristics (P), the example UA/FD process 2 102 usesEquation 7 for below for determining the audience characteristics (P).

$\begin{matrix}{{{minimize}\mspace{14mu} P},{{D\left( {P\text{:}Q} \right)} = {p_{k}\mspace{14mu} {\log \left( \frac{p_{k}}{q_{k}} \right)}}},{{{subject}\mspace{14mu} {to}\mspace{14mu} C_{P}P} = D_{P}}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

Since the example UA/FD process 2 102 involves a system of non-linearequations, the example processor resources 104 required to determine theunique audience is higher than the resources required to determine theunique audience using the example UA/FD process 1 101. Additionally,since the example UA/FD process 2 102 stores the audience/impressionmodel constraint matrix (C_(Q)), the audience/impressions totalconstraint vector (D_(Q)), the impressions characteristics vector (Q),and the audience characteristics (P), the amount of memory resources 105required for the example UA/FD process 2 102 is more than the amount ofthe example memory resources 105 required for the example UA/FD process1 101. However, as described above, the example UA/FD process 1 101cannot determine a unique audience based on logged impressionscorresponding to media accessed via one or more websites.

The example UA/FD process 3 103 requires the most processor resources,but less memory than the example UA/FD process 2 102. As the number ofwebsites associated with a request to log and/or the number of loggedimpressions per person per website increases, the amount of the examplememory resources 105 associated with the example UA/FD process 2 102becomes substantially large. In some examples, there is not enough ofthe example memory resources 105 to store all the values associated withthe example UA/FD process 2 102. In such examples, the UA/FD process 3103 may be used to determine the unique audience and/or frequencydistribution, because the UA/FD process 3 103 uses less memory todetermine the unique audience and/or frequency distribution for thelarge number of websites and/or logged impressions.

The example UA/FD process 3 103 decreases the amount of required memoryresources by calculating a combination of a set of the probabilities(e.g., to represent both the audience characteristics and the impressioncharacteristics) associated with the example UA/FD process 2 102,instead of calculating all of the audience characteristics and theimpression characteristics individually. For example, media exposuresfrom two websites may have billions of probabilities to represent theimpressions and audience characteristics, depending on the number ofexposures per person. The example UA/FD process 2 102 calculates andstores the billions of probabilities individually, while the exampleUA/FD process 3 103 only determines and stores four combinations.

The example UA/FD process 3 103 creates a combination matrix byenumerating all combinations that can occur, with each column being aconstraint corresponding to a webpage. In some examples, the constraintsmay be logged impressions corresponding to a first website, loggedimpressions corresponding to a second website, a total number ofimpressions, a total audience, etc. The example UA/FD process 3 103utilizes properties of the geometric series infinite summations tocreate a general formula for N websites, as shown in Equations 8 and 9below.

$\begin{matrix}{{{{impressions} \propto {\sum\limits_{i = 1}^{\infty}\; {\sum\limits_{j = 1}^{\infty}\; {\sum\limits_{k = 1}^{\infty}\; {z_{1}^{({a_{0} + {a_{1}i} + {a_{2}j} + {a_{3}k}})}z_{2}^{({b_{0} + {b_{1}i} + {b_{2}j} + {b_{3}k}})}}}}}} = \frac{z_{1}^{({a_{0} + a_{1} + a_{2} + a_{3}})}z_{2}^{({b_{0} + b_{1} + b_{2} + b_{3}})}}{\left( {1 - {z_{1}^{a_{1}}z_{2}^{b_{1}}}} \right)\left( {1 - {z_{1}^{a_{2}}z_{2}^{b_{2}}}} \right)^{2}\left( {1 - {z_{1}^{a_{3}}z_{2}^{b_{3}}}} \right)}},} & {{Equation}\mspace{14mu} 8} \\{{{{audience} \propto {\sum\limits_{i = 1}^{\infty}\; {\sum\limits_{j = 1}^{\infty}\; {\sum\limits_{k = 1}^{\infty}{j*\; z_{1}^{({a_{0} + {a_{1}i} + {a_{2}j} + {a_{3}k}})}z_{2}^{({b_{0} + {b_{1}i} + {b_{2}j} + {b_{3}k}})}}}}}} = \frac{z_{1}^{({a_{0} + a_{1} + a_{2} + a_{3}})}z_{2}^{({b_{0} + b_{1} + b_{2} + b_{3}})}}{\left( {1 - {z_{1}^{a_{1}}z_{2}^{b_{1}}}} \right)\left( {1 - {z_{1}^{a_{2}}z_{2}^{b_{2}}}} \right)^{2}\left( {1 - {z_{1}^{a_{3}}z_{2}^{b_{3}}}} \right)}},} & {{Equation}\mspace{14mu} 9}\end{matrix}$

where z_(i) is representative of the i^(th) constraint.

The example UA/FD process 3 103 creates a column based on a union oflonged impressions and the population. The example UA/FD process 3 103solves for N z values (e.g., exponents of LaGrangian multipliers foreach constraint used during optimization) to satisfy the N constraints.The example UA/FD process 3 103 calculates the N z values using a systemof non-linear equations. The example UA/FD process 3 103 modifies the zvalues corresponding to population constraints to solve for thepopulation constraints. Although the example UA/FD process 3 103 is themost computationally intensive UA/FD process requiring the mostprocessor resources, it requires less of the example memory resources105 than the example UA/FD process 2 102.

FIG. 1A illustrates the example processor resources 104 and the examplememory resources 105 required to estimate a unique audience and/orfrequency distribution corresponding to logged impressions in theexample situations 110, 115, 120. In the example situations 110, 115,120, an AME determines a unique audience based on impressions logged bya database proprietor for media 100 presented to numerous audiencemembers via one webpage 106. Additionally, the example situations 110,115, 120 include a small number of impressions per viewer (e.g., lessthan 10 impressions per person). In some examples, the AME may receive atotal number of impressions associated with the example media 100 (e.g.,a total number of impressions logged for the media 100). Additionally,the AME may receive aggregate database proprietor impression data from adatabase proprietor based on the media 100. In examples disclosedherein, aggregate database proprietor impression data is a reporting ofprocessed impression totals and other metrics based on impressionslogged by the database proprietor for numerous audience members exposedto the media 100. For example, the database proprietor may generate theaggregate database proprietor impression data by tallying, averaging,de-duplicating and/or performing any other mathematical and/or filteringoperations on database proprietor demographic impressions logged by thedatabase proprietor for the media 100 exposed to numerous audiencemembers. In some examples, the database proprietor may also generateaggregate database proprietor impression data by associating impressionmetrics with demographic groups. In some examples, the aggregatedatabase proprietor impression data includes a total number of uniquepeople exposed to the example media 100 that are registered sisubscribers of the database proprietor (e.g., a partial audiencecorresponding to registered database proprietor users) and a totalnumber of database proprietor demographic impressions corresponding tothe partial audience.

As shown in the example situation 110, when the example UA/FD process 1101 calculates the unique audience for logged impressions associatedwith the media 100 from the example website 106 with a small number ofimpressions per viewer, the example processor resources 104 and theexample memory resources 105 used for such calculations are low. Asshown in the example situation 115, the example UA/FD process 2 102requires more of the example processor resources 104 than the exampleUA/FD process 1 101 and less of the example resources 104 than theexample UA/FD process 3 103. Additionally, the example memory resources105 needed for the example UA/FD process 2 102 remain relatively low. Asshown in the example situation 120, the example UA/FD process 3 103 usesmore of the example processor resources 104 than the UA/FD process 1 101and the UA/FD process 2 102, but the example memory resources 105 remainrelatively low. In the illustrated example of FIG. 1A, the example UA/FDprocess 1 101 is the optimal UA/FD process to use relative to the UA/FDprocess 2 102 and the UA/FD process 3 103 because it uses less of boththe example processor resources 104 and the example memory resources 105than used by the example process 2 102 and the example process 3 103.

FIG. 1B illustrates the example processor resources 104 and the examplememory resources 105 required to estimate a unique audience and/orfrequency distribution in the example situations 125, 130, 135. In theexample situations 125, 130, 135, an AME determines unique audiencescorresponding to impressions logged by a database proprietor for theexample media 100 presented to numerous audience members via the examplesmall number of websites 108 (e.g., more than one but less than athreshold number (5)). The unique audiences may include unique audiencesfor each website of the small number of websites 108 as well as a totalunique audience for all the websites 108. Additionally, the examplesituations 125, 130, 135 include a small number of impressions perviewer (e.g., 2-10 impressions per viewer). In some examples, the AMEmay receive a total number of logged impressions associated with themedia 100. Additionally, the AME may receive aggregate databaseproprietor impression data from one or more database proprietors basedon the media 100. In some examples, the aggregate database proprietorimpression data includes a total number unique people exposed to themedia 100 that are registered to the one or more database proprietors(e.g., the partial audience corresponding to registered databaseproprietor users) and a total number of logged database proprietordemographic impressions corresponding to the partial audience.

In the example situation 125, the example UA/FD process 1 101 cannot beused determine a unique audience for each of the example small number ofwebpages 108 (e.g., a unique audience for website A, a unique audiencefor website B, etc.). That is, the example UA/FD process 1 101 isconfigured to determine a unique audience for a single website such asthe website 106 of FIG. 1A, but not for numerous websites. As shown inthe example situation 130, the example UA/FD process 2 102 requires lessof the example processor resources 104 than the example UA/FD process 3103 selected at situation 135. Additionally, the example memoryresources 105 required for the example UA/FD process 2 102 is more thanthe example memory resources 105 associated with the example UA/FDprocess 3 103 selected at situation 135. As shown in the examplesituation 135, the example UA/FD process 3 103 requires more of theexample processor resources 104 than in the example process 2 102.Additionally, the required example memory resources 105 remainsrelatively lower than in the UA/FD process 2 102. In the illustratedexample of FIG. 1B, the determination of which UA/FD process is optimalis based on the available processor resources 104 and/or the examplememory resources 105 of a computer (e.g., the processor system 1000 ofFIG. 10). For example, in a system where the available memory resources105 are low, the optimal process may be a process that requires lessmemory resources 105. In some examples, the example processor resources104 and/or the example memory resources 105 are weighted based on userand/or manufacture preferences to determine which UA/FD process isoptimal for particular circumstances of a number of websites and anumber of impressions. For example, if the system operating the UA/FDprocesses has a small amount of the example processor resources 104 buta large amount of the example memory resources 105, a user may give moreweight to the processor resources 104 in order to select a process thatuses more processor resources 104.

FIG. 1C illustrates the example processor resources 104 and the examplememory resources 105 required to estimate a unique audience and/orfrequency distribution in the example situations 140, 145, 150. Theexample situations 140, 145, 150 may require an AME to determine aunique audience based on impressions logged by a database proprietor forthe example media 100 presented to numerous audience members via theexample large number of websites 109 (e.g., more than a thresholdnumber). Additionally, the example situations 140, 145, 150 include alarge number of impressions per viewer (e.g., more than 10). In someexamples, the AME may receive a total number of media impressions loggedfor the media 100. Additionally, the AME may receive aggregate databaseproprietor impression data from one or more database proprietors basedon the media 100. In some examples, the aggregate database proprietorimpression data includes a total number of unique people exposed to themedia 100 that are registered subscribers of the database proprietor(e.g., the partial audience corresponding to registered databaseproprietor users) and a total number database proprietor demographicimpressions corresponding to the partial audience.

In the example situation 140, the example UA/FD process 1 101 cannot beused to determine a unique audience for each of the example large numberof websites 109 (e.g., a unique audience for website A, a uniqueaudience for website B, etc.). That is, the example UA/FD process 1 101is configured to determine a unique audience for a single website suchas the website 106 of FIG. 1A, but not for numerous websites. As shownin the example situation 145, the example UA/FD process 2 102 requiresless of the example processor resources 104 than the example UA/FDprocess 3 103 selected at situation 150. Additionally, the examplememory resources 105 usage for the example UA/FD process 2 102 selectedat situation 145 is more than the example memory resources 105associated with the example UA/FD process 3 103 selected at situation150. In some examples, there may not be enough of the example memoryresources 105 to determine a solution (e.g., when the number of websitesand/or the number of impressions per view are sufficiently large) usingthe UA/FD process 2 102. In such examples the unique audience cannot bedetermined using the example UA/FD process 2 102. As shown in theexample situation 150, the example UA/FD process 3 103 requires more ofthe example processor resources 104 and less of the example memoryresources 105 than the example process 2 102 selected at situation 145.In the illustrated example of FIG. 1C, the determination of which UA/FDprocess is optimal is based on the available processor resources 104and/or the example memory resources 105. If, as described above, thereis not enough of the example memory resources 105 to determine asolution using the example UA/FD process 2 102, a user may give moreweight to the processor resources 104 in order to select a process thatuses more processor resources 104.

FIG. 2 illustrates example client devices 202 that report audienceimpression requests for Internet-based media (e.g., the media 100 ofFIGS. 1A-1C) to impression collection entities 208 to identify a uniqueaudience and/or a frequency distribution for the Internet-based media.The illustrated example of FIG. 2 includes the example client devices202, an example network 204, example impression requests 206, and theexample impression collection entities 208. As used herein, animpression collection entity 208 refers to any entity that collectsimpression data such as, for example, an example AME 212 and/or anexample database proprietor 210. In the illustrated example, the AME 212includes an example audience/impression determiner 214.

The example client devices 202 of the illustrated example may be anydevice capable of accessing media over a network (e.g., the examplenetwork 204). For example, the client devices 202 may be an examplemobile device 202 a, an example computer 202 b, 202 d, an example tablet202 c, an example smart television 202 e, and/or any otherInternet-capable device or appliance. Examples disclosed herein may beused to collect impression information for any type of media includingcontent and/or advertisements. Media may include advertising and/orcontent delivered via websites, streaming video, streaming audio,Internet protocol television (IPTV), movies, television, radio and/orany other vehicle for delivering media. In some examples, media includesuser-generated media that is, for example, uploaded to media uploadsites, such as YouTube, and subsequently downloaded and/or streamed byone or more other client devices for playback. Media may also includeadvertisements. Advertisements are typically distributed with content(e.g., programming). Traditionally, content is provided at little or nocost to the audience because it is subsidized by advertisers that pay tohave their advertisements distributed with the content. As used herein,“media” refers collectively and/or individually to content and/oradvertisement(s).

The example network 204 is a communications network. The example network204 allows the example impression requests 206 from the example clientdevices 202 to the example impression collection entities 208. Theexample network 204 may be a local area network, a wide area network,the Internet, a cloud, or any other type of communications network.

The impression requests 206 of the illustrated example includeinformation about accesses to media at the corresponding client devices202 generating the impression requests. Such impression requests 206allow monitoring entities, such as the impression collection entities208, to collect a number of media impressions for different mediaaccessed via the client devices 202. By collecting media impressions,the impression collection entities 204 can generate media impressionquantities for different media (e.g., different content and/oradvertisement campaigns).

The impression collection entities 208 of the illustrated exampleinclude the example database proprietor 210 and the example AME 212. Inthe illustrated example, the example database proprietor 210 may be oneof many database proprietors that operate on the Internet to provideservices to subscribers. Such services may be email services, socialnetworking services, news media services, cloud storage services,streaming music services, streaming video services, online retailshopping services, credit monitoring services, etc. Example databaseproprietors include social network sites (e.g., Facebook, Twitter,MySpace, etc.), multi-service sites (e.g., Yahoo!, Google, Axiom,Catalina, etc.), online retailer sites (e.g., Amazon.com, Buy.com,etc.), credit reporting sites (e.g., Experian), streaming media sites(e.g., YouTube, etc.), and/or any other site that maintains userregistration records.

In some examples, execution of the beacon instructions corresponding tothe media 100 causes the client devices 202 to send impression requests206 to servers 211, 213 (e.g., accessible via an Internet protocol (IP)address or uniform resource locator (URL)) of the impression collectionentities 208 in the impression requests 206. In some examples, thebeacon instructions cause the client devices 202 to locate device and/orusers identifiers and media identifiers in the impression requests 206.The device/users identifier may be any identifier used to associatedemographic information with a user or users of the client devices 202.Example device/user identifiers include cookies, hardware identifiers(e.g., an international mobile equipment identity (IMEI), a mobileequipment identifier (MEID), a media access control (MAC) address,etc.), an app store identifier (e.g., a Google Android ID, an Apple ID,an Amazon ID, etc.), an open source unique device identifier (OpenUDID),an open device identification number (ODIN), a login identifier (e.g., ausername), an email address, user agent data (e.g., application type,operating system, software vendor, software revision, etc.), an Ad ID(e.g., an advertising ID introduced by Apple, Inc. for uniquelyidentifying mobile devices for purposes of serving advertising to suchmobile devices), third-party service identifiers (e.g., advertisingservice identifiers, device usage analytics service identifiers,demographics collection service identifiers), etc. In some examples,fewer or more device/user identifier(s) 228 may be used. The mediaidentifiers (e.g., embedded identifiers, embedded codes, embeddedinformation, signatures, etc.) enable the impression collection entities208 can identify to media (e.g., the media 100) objects accessed via theclient devices 202. The impression requests 206 of the illustratedexample cause the AME 212 and/or the database proprietor 210 to logimpressions for the media 100. In the illustrated example, an impressionrequest is a reporting to the AME 202 and/or the database proprietor 210of an occurrence of the media 100 being presented at the client device202. The impression requests 206 may be implemented as a hypertexttransfer protocol (HTTP) request. However, whereas a transmitted HTTPrequest identifies a webpage or other resource to be downloaded, theimpression requests 206 include audience measurement information (e.g.,media identifiers and device/user identifier) as its payload. The server211, 213 to which the impression requests 206 are directed is programmedto log the audience measurement information of the impression requests206 as an impression (e.g., a media impression such as advertisementand/or content impressions depending on the nature of the media accessedvia the client device 202). In some examples, the server 211, 213 of thedatabase proprietor 201 or the AME 212 may transmit a response based onreceiving an impression request 206. However, a response to theimpression request 206 is not necessary. It is sufficient for the server211, 213 to receive the impression request 206 to log an impressionrequest 206.

The example database proprietor 210 maintains user account recordscorresponding to users registered for services (such as Internet-basedservices) provided by the database proprietors. That is, in exchange forthe provision of services, subscribers register with the databaseproprietor 210. As part of this registration, the subscribers providedetailed demographic information to the database proprietor 210.Demographic information may include, for example, gender, age,ethnicity, income, home location, education level, occupation, etc. Inthe illustrated example, the database proprietor 210 sets a device/useridentifier on a subscriber's client device 202 that enables the databaseproprietor 210 to identify the subscriber.

In the illustrated example, the example AME 212 does not provide themedia 100 to the client devices 202 and is a trusted (e.g., neutral)third party (e.g., The Nielsen Company, LLC) for providing accuratemedia access (e.g., exposure) statistics. The example AME 212 includesthe example audience/impressions determiner 214. As further disclosedherein, the example audience/impressions determiner 214 provides mediaaccess statistics related to the example impression requests 206. Insome examples, the audience/impressions determiner 214 calculates atotal reach (e.g., a total unique audience) exposed to particular media(e.g., the media 100) based on the example impression requests 206 anddata from the example database proprietor 210 (e.g., database proprietordemographic impressions and/or partial audience). Additionally oralternatively, the example audience/impressions determiner 214calculates a frequency distribution indicative of (1) a total quantityof unique audience members who have not been exposed to a particularmedia, (2) a total quantity of unique audience members who have beenexposed to the particular media exactly once, (3) a total quantity ofunique audience members who have been exposed to the particular mediaexactly twice, etc. Additionally, the example audience/impressionsdeterminer 214 may calculate any statistic related to the exampleimpression requests 206. As disclosed herein, the exampleaudience/impressions determiner 214 determines an optimal UA/FD process(e.g., one of the UA/FD processes 101, 102, 103 of FIGS. 1A-1C) todetermine one or more unique audiences corresponding to one or morewebsites outputting the media and/or other media impression data basedon input data and/or operator preferences. The optimal UA/FD process maybe determined to optimize usage of resources (e.g., processor resourcesand/or memory).

In operation, the example client devices 202 employ web browsers and/orapplications (e.g., apps) to access media. Some of the web browsers,applications, and/or media include instructions that cause the exampleclient devices 202 to report media monitoring information to one or moreof the example impression collection entities 208. That is, when theclient device 202 of the illustrated example accesses media, a webbrowser and/or application of the client device 202 executesinstructions in the media, in the web browser, and/or in the applicationto send the example impression request 206 to one or more of the exampleimpression collection entities 208 via the example network 206. Theexample impression requests 206 of the illustrated example includeinformation about accesses to the media 100 and/or any other media atthe corresponding client devices 202 generating the impression requests206. Such impression requests allow monitoring entities, such as theexample impression collection entities 208, to collect media impressionsfor different media accessed via the example client devices 202. In thismanner, the impression collection entities 208 can generate mediaimpression quantities for different media (e.g., different contentand/or advertisement campaigns).

When the example database proprietor 210 receives the example impressionrequest 206 from the example client device 202, the example databaseproprietor 210 requests the client device 202 to provide a device/useridentifier that the database proprietor 210 had previously set for theexample client device 202. The example database proprietor 210 uses thedevice/user identifier corresponding to the example client device 202 toidentify the subscriber of the client device 202.

In the illustrated example, three of the client devices 202 a, 202 b,and 202 c have DP IDs (DP device/user IDs) that identify correspondingsubscribers of the database proprietor 210. In this manner, when theclient devices 202 a, 202 b, 202 c corresponding to subscribers of theexample database proprietor 210 send impression requests 206 to theimpression collection entities 208, the database proprietor 210 mayrecord database proprietor demographic impressions for the user. In theillustrated example, the client devices 202 d, 202 e do not have DP IDs.As such, the example database proprietor 210 is unable to identify theclient devices 202 d, 202 e due to those client devices not having DPIDs set by the example database proprietor 210. The client devices 202d, 202 e may not have DP IDs set by the database proprietor 210 if, forexample, the client devices 202 d, 202 e do not accept cookies, a userdoes not have an account with the database proprietor 210 or the userhas an account with the database proprietor 210 but has cleared the DPID (e.g., cleared a cookie cache) and deleted the database proprietor'sDP ID before or at the time of a media exposure. In such instances, ifthe user device 202 is, for example, redirected to contact the databaseproprietor 210 using the system disclosed in Mainak et al., U.S. Pat.No. 8,370,489, the database proprietor 210 is not able to detectdemographics corresponding to the media exposure and, thus, does notreport/log any audience or database proprietor demographic impressionsfor that exposure. In examples disclosed herein, the client devices 202d, 202 e are referred to herein as client devices over which thedatabase proprietor 210 has non-coverage because the database proprietor210 is unable to identify demographics corresponding to those clientdevices 202 d, 202 e. As a result of the non-coverage, the databaseproprietor 210 underestimates the audience size and number of mediaimpressions for corresponding media accessed via the client devices 202when, for example, operating within the system of Mainak et al., U.S.Pat. No. 8,370,489.

The example AME 212 receives database proprietor demographic impressiondata from the example database proprietor 210. The database proprietordemographic impression data may include information relating to a totalnumber of the logged database proprietor demographic impressions thatcorrespond with a registered user of the database proprietor 210, atotal number of registered users (e.g., a partial audience) that wereexposed to media associated with the logged database proprietordemographic impressions, and/or any other information related to thelogged database proprietor demographic impressions (e.g., demographics,a total number of registered users exposed to the media 100 more thanonce, etc.). The example audience/impressions determiner 214 determinesa total number of logged media impressions (including but not limited tothe number of logged database proprietor demographic impressions fromthe example database proprietor 210) and a total unique audience basedon impression requests 206, survey data, census data, and/or data from amedia provider. As further disclosed herein, the exampleaudience/impression determiner 214 determines impression statisticsbased on the database proprietor demographic impressions associated withthe example impression requests 206 and the database proprietordemographic impression data from the example database proprietor 210.The example audience/impression determiner 214 determines a uniqueaudience(s) and/or frequency distribution using at least one of threeUA/FD processes the example UA/FD process 1 101, the example UA/FDprocess 1 102, and the example UA/FD process 3 103 of FIGS. 1A-1C),depending on a desired number of logged impressions per user and anumber of websites associated with the logged media impressions.

FIG. 3 is a block diagram of the example audience/impression determiner214 of FIG. 2, disclosed herein, to determine audience measurement datarelated to the example impression requests 206 (FIG. 2) and the exampledatabase proprietor 210 (FIG. 2). While the example audience/impressiondeterminer 214 (FIG. 2) is described in conjunction with the exampleclient devices 202 (FIG. 2) and the example impression collectionentities 208 (FIG. 2), the example audience/impression determiner 214may be utilized to determine impression data based on any type ofcomputing device and/or collection entity. The exampleaudience/impression determiner 214 includes an example data interface300, an example process selector 302, an example audience datacalculator 304, an example impression data calculator 306, and anexample report generator 308.

The example data interface 300 receives the example impression requests206 and data from the example database proprietor 303 (e.g., demographicdatabase proprietor impression data). Initially, the example datainterface 300 receives an impression request 206 to log an impression.The impression request 206 of the illustrated example includes a mediaidentifier (ID) 301 to identify the example media 100. The media ID 301is used to monitor impressions of media 100 and aggregate databaseproprietor impression data (e.g., database proprietor demographicimpressions and/or a partial audience) associated with the media 100.

The example process selector 302 processes the received databaseproprietor demographic impression data from the example databaseproprietor 303 to select an optimal UA/FD process (e.g., one of theexample UA/FA processes 101, 102, 103 of FIGS. 1A-1C) and/or a frequencydistribution for determining a total unique audience and/or a frequencydistribution for the example media 100. In some examples, the databaseproprietor demographic impression data may include data describingdemographics of client device users (e.g., total unique 18-25 year oldmales exposed to the media 100). Additionally, the database proprietordemographic impression data may include (1) how many client device userswere exposed to the media 100 exactly once, exactly twice, etc., (2) howmany client device users were exposed to the media 100 from a firstwebsite, a second website, etc. (3) and/or more complex combinations ofmedia exposure (e.g., how many high income males were exposed to themedia impression exactly three times from both website “A” and website“B”). In some examples, the example process selector 302 selects a UA/FDprocess based on the number of inputs (e.g., the number of websites)and/or the complexity and/or accuracy of a desired output. For example,the example UA/FD process 101 is optimal (e.g., optimizing use of theexample processor resources 104 and/or the example memory resources 105of FIG. 1A) to determine a total unique audience based on mediaimpressions from one website (e.g., the example website 106 of FIG. 1A)due to its relatively low use of the processor resources 104 and thememory resources 105, whereas the UA/FD process 101 may be least optimalor insufficient to determine a total unique audience based on mediaimpressions for the media 100 accessed via the multiple websites (e.g.,the websites 108, 109) since the UA/FD process 101 cannot determineunique audiences and frequency distributions for more than one website.In some examples, the process selector 302 may estimate the amount ofthe example processor resources 104 and/or the amount of example memoryresources 105 required to perform each of the example UA/FD processesbased on the given inputs (e.g., number of websites, number ofimpressions per person). In such examples, the process selector 302 mayassign different weights to the example processor resources 104 and/orthe example memory resources 105 required by each UA/FD process todetermine the optimal UA/FD process. The weights may be based on userand/or manufacturer preferences. For example, if a system has limitedmemory resources (e.g., such as the memory resources 105 of FIGS.1A-1C), the user may adjust the weights so that the UA/FD processes thatrequire less memory resources are optimal.

The example audience data calculator 304 calculates a unique audiencebased on the received database proprietor demographic impression data303 and the selected UA/FD process. In some examples, the audience datacalculator 304 inputs data related to the logged media impressionsand/or the aggregate database proprietor impression data into a formulato calculate the unique audience. In some examples, the audience datacalculator 304 creates a population model constraint matrix (e.g., theexample constraint matrix 825 of FIG. 8C) and a population totalconstraint vector (e.g., the example population constraint vector 827 ofFIG. 8C) to calculate the unique audience. In some examples, theaudience data calculator 304 solves various non-linear systems ofequations to calculate the unique audience(s). The calculation of theunique audience and/or various statistics is further described in FIGS.5-7.

The example impression data calculator 306 calculates a frequencydistribution based on the received data 303 and the selected UA/FDprocess. In some examples, the impression data calculator 306 inputsdata related to the logged media impressions and/or the aggregatedatabase proprietor impression data into a formula to calculate thefrequency distribution. In some examples, the impression data calculator306 creates an audience/impressions model constraint matrix (C_(Q)) andan audience/impressions total constraint vector (D_(Q)) to calculate thefrequency distribution as described above in connection with Equation 6.In some examples, the impression data calculator 306 solves variousnon-linear systems of equations (e.g., as described above in connectionwith Equations 8 and 9) to calculate the frequency distribution. Thecalculation of the frequency distribution and/or various statistics isfurther described below in connection with FIGS. 5-7.

The example report generator 308 generates reports based on the variousstatistics calculated by the example audience data calculator 304 andthe example impression data calculator 306. In some examples, the reportgenerated by the example report generator 308 includes a uniqueaudience(s) and/or a frequency distribution. In some examples, thereport generated by the example report generator 308 includesdemographic data (e.g., a unique audience and/or frequency distributionfor a particular demographic). In some examples, the report generator308 includes data for the one or more websites that are associated withan impression associated with the example media 100. In some examples,the report generator 308 includes data in reports that describe how manypeople within the unique audience were exposed to the media 100 exactlyonce, twice, etc. In some examples, the report generator 308 combinesdata from logged impressions of the media 100 and/or other mediaassociated with a particular company. For example, the report generatedby the example report generator 308 may combine data indicating a totalaudience for three distinct advertisements for a particular company.Such reports may include data indicating how many people saw at leastone of the three advertisements, how many people saw two of the threeadvertisements from website A or B, how many people saw the first twoadvertisements, but missed the last advertisement, etc. In someexamples, the report generator 308 credits media associated with loggedimpressions based on the unique audience and/or frequency distribution.

While example manners of implementing the example audience/impressionsdeterminer 214 of FIG. 2 are illustrated in FIG. 3, elements, processesand/or devices illustrated in FIG. 3 may be combined, divided,re-arranged, omitted, eliminated and/or implemented in any other way.Further, the example data interface 300, the example process selector302, the example audience data calculator 304, the impression datacalculator 306, the example report generator 308, and/or, moregenerally, the example audience/impressions determiner 214 of FIG. 3 maybe implemented by hardware, machine readable instructions, software,firmware and/or any combination of hardware, machine readableinstructions, software and/or firmware. Thus, for example, any of theexample data interface 300, the example process selector 302, theexample audience data calculator 304, the impression data calculator306, the example report generator 308, and/or, more generally, theexample audience/impressions determiner 214 of FIG. 3 could beimplemented by analog and/or digital circuit(s), logic circuit(s),programmable processor(s), application specific integrated circuit(s)(ASIC(s)), programmable logic device(s) (PLD(s)) and/or fieldprogrammable logic device(s) (FPLD(s)). When reading any of theapparatus or system claims of this patent to cover a purely softwareand/or firmware implementation, at least one of the example datainterface 300, the example process selector 302, the example audiencedata calculator 304, the impression data calculator 306, the examplereport generator 308, and/or, more generally, the exampleaudience/impressions determiner 214 of FIG. 3 is/are hereby expresslydefined to include a tangible computer readable storage device orstorage disk such as a memory, a digital versatile disk (DVD), a compactdisk (CD), a Blu-ray disk, etc. storing the software and/or firmware.Further still, the example audience/impression determiner 214 of FIG. 3include elements, processes and/or devices in addition to, or insteadof, those illustrated in FIGS. 4-7, and/or may include more than one ofany or all of the illustrated elements, processes and devices.

Flowcharts representative of example machine readable instructions forimplementing the example audience/impression determiner 214 of FIG. 3are shown in FIGS. 4-7. In the examples, the machine readableinstructions comprise a program for execution by a processor such as theprocessor 1012 shown in the example processor platform 1000 discussedbelow in connection with FIG. 10. The program may be embodied in machinereadable instructions stored on a tangible computer readable storagemedium such as a CD-ROM, a floppy disk, a hard drive, a digitalversatile disk (DVD), a Blu-ray disk, or a memory associated with theprocessor 1012, but the entire program and/or parts thereof couldalternatively be executed by a device other than the processor 1012and/or embodied in firmware or dedicated hardware. Further, although theexample program is described with reference to the flowchartsillustrated in FIGS. 4-7, many other methods of implementing the exampleaudience/impression determiner 214 of FIG. 3 may alternatively be used.For example, the order of execution of the blocks may be changed, and/orsome of the blocks described may be changed, eliminated, or combined.Although the flowchart of FIG. 4 depicts example operations in anillustrated order, these operations are not exhaustive and are notlimited to the illustrated order. In addition, various changes andmodifications may be made by one skilled in the art within the spiritand scope of the disclosure. For example, blocks illustrated in theflowchart may be performed in an alternative order or may be performedin parallel.

As mentioned above, the example processes of FIGS. 4-7 may beimplemented using coded instructions (e.g., computer and/or machinereadable instructions) stored on a tangible computer readable storagemedium such as a hard disk drive, a flash memory, a read-only memory(ROM), a compact disk (CD), a digital versatile disk (DVD), a cache, arandom-access memory (RAM) and/or any other storage device or storagedisk in which information is stored for any duration (e.g., for extendedtime periods, permanently, for brief instances, for temporarilybuffering, and/or for caching of the information). As used herein, theterm tangible computer readable storage medium is expressly defined toinclude any type of computer readable storage device and/or storage diskand to exclude propagating signals and to exclude transmission media. Asused herein, “tangible computer readable storage medium” and “tangiblemachine readable storage medium” are used interchangeably. Additionallyor alternatively, the example processes of FIGS. 4-7 may be implementedusing coded instructions (e.g., computer and/or machine readableinstructions) stored on a non-transitory computer and/or machinereadable medium such as a hard disk drive, a flash memory, a read-onlymemory, a compact disk, a digital versatile disk, a cache, arandom-access memory and/or any other storage device or storage disk inwhich information is stored for any duration (e.g., for extended timeperiods, permanently, for brief instances, for temporarily buffering,and/or for caching of the information). As used herein, the termnon-transitory computer readable medium is expressly defined to includeany type of computer readable storage device and/or storage disk and toexclude propagating signals and to exclude transmission media. As usedherein, when the phrase “at least” is used as the transition term in apreamble of a claim, it is open-ended in the same manner as the term“comprising” is open ended. In addition, the term “including” isopen-ended in the same manner as the term “comprising” is open-ended.

The example machine readable instructions 400 illustrated in FIG. 4 maybe executed to implement the audience/impressions determiner 214 ofFIGS. 2 and 3 to select a UA/FD process (e.g., the UA/FD processes 101,102, 103 of FIGS. 1A-1C) that optimizes the example processor resources104 and/or the example memory resources 105 of FIG. 1 for use indetermining unique audience(s) and/or frequency distribution of theunique audience based on logged impressions corresponding to the media100 (FIGS. 1A-1C and 2).

Initially, at block 402 of FIG. 4, the example data interface 300 (FIG.3) receives instructions to measure media exposure corresponding to themedia 100 of FIG. 1. For example, a company corresponding to the media100 may request the AME 212 (FIG. 2) to measure media exposure for themedia 100 accessed via one website (e.g., the website 106 of FIG. 1A).Alternatively, the company may request the AME 212 to measure mediaexposure for the media 100 accessed via two or more websites (e.g., theexample websites 108, 109 of FIGS. 1B-1C). In some examples, theinstructions include a total number of logged media impressions and/or atotal universe audience size.

At block 404 of FIG. 4, the example process selector 302 (FIG. 3)determines if the instructions of block 402 to measure media exposureare for media impressions accessed via one website. For example, theinstructions may request a total unique audience for the media 100 basedon media impressions accessed via only one website (e.g., the examplewebsite 106). If the example processor selector 302 determines that theinstructions of block 402 are to measure media exposure based on mediaimpressions accessed via one website, the example process selector 302selects the example UA/FD process 1 101 of FIG. 1.

If the example processor selector 302 determines that the instructionsof blocks 402 are not for measuring media exposure based on mediaimpressions accessed via one website (e.g., the request is based on thelogged impressions from the example two or more websites 108, 109), theexample selector 302 determines if the number of probabilitiesassociated with the numerous websites and impressions per person is morethan a threshold amount (block 406). For example, as described above, asthe number of websites and/or impressions per person increases, theamount of possible probabilities to solve and store increases. In someexamples, a computer to execute the selected UA/FD process does not haveenough of the required example memory resources 105 to store all of thepossible probabilities associated with the websites and/or impressions.Since a person can view the media 100 a large number of times, thenumber of possible probabilities becomes nearly infinite. Thus, theexample UA/FD process 2 102 (FIGS. 1A-1C) usually limits the number ofimpressions per person in order to determine a unique audience based onthe limited number of impressions per person. The number ofprobabilities is equivalent to the number of impressions per person tothe power of the number of websites. For example, if the request to logimpressions is based on the media 100 from two websites the examplewebsites 108, 109) where the most a client device can report animpression associated with the media 100 (e.g., be exposed to the media)is five times, then the number of probabilities is twenty-five (e.g.,5²=25) due to the capacity of mathematics associated with the UA/FDprocess modeling.

If the number of probabilities is not more than the threshold amount atblock 406, then the example process selector 302 selects the exampleUA/FD process 2 102 (FIGS. 1A-1C). If the number of probabilities ismore than the threshold amount, the example process selector 302 selectsthe example UA/FD process 3 103 (FIGS. 1A-1C). The threshold amount maybe based on AME manufacturer settings. For example, in systems that havelimited memory resources, the example AME 212 of FIG. 2 may select asmaller threshold amount for block 406 for a system with a large amountof memory resources. In some examples, the amount of example processorresources 104 and/or the amount of the example memory resources 105 areweighted to determine the threshold amount.

Turning now to FIG. 5A, the example machine readable instructionsillustrated in FIG. 5A may be executed to cause the audience/impressionsdeterminer 214 of FIG. 3 to determine a total unique audience andfrequency distribution using the example UA/FD process 1 101.

FIG. 5A is an example flowchart 500 representative of example machinereadable instructions that may be executed to implement the exampleaudience/impressions determiner 214 of FIGS. 2 and 3 to determine atotal unique audience (X) and a frequency distribution of the uniqueaudience based on the impressions of the example media 100 using theexample UA/FD process 1 101 (FIG. 1A).

At block 502 of FIG. 5A, the example data interface 300 receives loggedimpressions from a plurality of the client devices 202 in a monitored ormeasured region. For example, there may be 1,000 total loggedimpressions (T) associated with the media 100 from a population of50,000 people in a monitored region (e.g., a universe) (U). For example,the universe is a monitored region such as a state, a country, acontinent, the world, etc. In some examples, the data interface 300receives impression requests at the server 213 (FIG. 2) of the AME 212(FIG. 2) from the client devices 202 (FIG. 2) via the network 204 (FIG.2). In such examples, the server 213 logs the impressions correspondingto the example media 100 (FIG. 2) and determines a quantity of uniquepeople identified by the example database proprietor 210 (e.g., apartial audience corresponding to registered database proprietor users)associated with the media 100 using the example server 213. As describedabove, not everyone in the universe population viewed the media 100 andsome of the 1,000 total logged impressions were presented to the sameclient device 202. Because at least one person in the universepopulation may have viewed the media 100 more than once, the totalnumber of the logged impressions usually is not representative of aunique audience of the media 100. As described above, the example UA/FDprocess 1 101 is based on an equation derived from the principles ofmaximum entropy and minimum cross entropy. The equation can besimplified (e.g., in order to use less of the example processorresources 104) based on the sizes of the universe population, the numberof the logged impressions corresponding the database proprietor 210(e.g., database proprietor demographic impressions), the unique audienceregistered to the database proprietor 210 associated with the loggedimpressions, and/or the total number of the logged impressions (e.g.,media impressions).

At block 504 of FIG. 5A, the example data interface 300 (FIG. 3) obtainsa number of logged demographic database proprietor impressions and atotal number of registered users identified by the database proprietor210 (FIG. 2). In the illustrated example, for the monitored media 100,the database proprietor 210 may identify a count of 600 loggedimpressions (R) that correspond to a count of 200 unique people (A) thatare registered users of the database proprietor 210.

At block 506 of FIG. 5A, the example audience data calculator 304 (FIG.3) determines if the universe population is more than a first thresholdsize. The first threshold size may be based on a user and/ormanufacturer preference. For example, the threshold size may be based ona desired precision of the results (e.g., the higher the threshold themore precise the results). If the universe population is not more thanthe first threshold size at block 506, then the audience data calculator304 determines a unique audience for the media 100 (e.g., theadvertisement) based on an example first unique audience sub-process(e.g., herein referred to as unique audience sub-process 1.1) (block508). The example unique audience sub-process 1.1 is represented as

$\frac{\left( {R - A} \right)\left( {U - A} \right)}{A^{2}} = {\frac{\left( {T - X} \right)\left( {U - X} \right)}{X^{2}}.}$

An example manner of implementing the example unique audiencesub-process 1.1 is described below in connection with FIG. 5B. Usingunique audience sub-process 1.1, the audience data calculator 304determines the unique audience (X) to be 270 people

$\left( {{e.g.},{\frac{\left( {600 - 200} \right)\left( {50000 - 200} \right)}{200^{2}} = \frac{\left( {1000 - X} \right)\left( {50000 - X} \right)}{X^{2}}}} \right).$

If the universe population is more than the first threshold, uniqueaudience sub-process 1.1 can be simplified. That is, when the universeis sufficiently large the unique audience sub-process 1.1 is simplifiedby taking the limit of the unique audience sub-process 1.1 as U goes toinfinity, as further described below.

If the example audience data calculator 304 determines at block 506 thatthe universe population is more than the threshold size, controladvances to block 510. At block 510, the example audience datacalculator 304 determines if there is a difference between the number oflogged impressions and audience size less than a second threshold. Thesecond threshold size may be based on a user and/or manufacturerpreference. For example, the threshold size may be based on a desiredprecision of the results (e.g., the higher the threshold the moreprecise the results). If at block 510 the difference between the numberof logged impressions (R and T) and the audience sizes (A and X) is lessthan the second threshold, the example audience data calculator 304determines a unique audience (X) based on an example second uniqueaudience sub-process (e.g., herein referred to as unique audiencesub-process 1.2) (block 512). An example manner of implementing theexample unique audience sub-process 1.2 is described below in connectionwith FIG. 5C. The example, unique audience sub-process 1.2 isrepresented as

$\frac{A^{2}}{R - A} = {\frac{X^{2}}{T - X}.}$

Unique audience sub-process 1.2 is based on a mathematical rearrangementof unique audience sub-process 1.1 since the universe population islarge

$\left( {{e.g.},{{\lim\limits_{U\rightarrow\infty}\frac{\left( {R - A} \right)\left( {U - A} \right)}{A^{2}}} = {{\frac{\left( {T - X} \right)\left( {U - X} \right)}{X^{2}}\overset{yields}{\rightarrow}\frac{\left( {R - A} \right)}{A^{2}}} = {{\frac{\left( {T - X} \right)}{X^{2}}\mspace{14mu} {or}\mspace{14mu} \frac{A^{2}}{R - A}} = \frac{X^{2}}{T - X}}}}} \right).$

Using the unique audience sub-process 1.2, the audience data calculator304 determines the unique audience (X) to be 270

$\left( {{e.g.},{\frac{200^{2}}{600 - 200} = \frac{X^{2}}{1000 - X}}} \right).$

If at block 510 the difference between the number of logged impressions(R and T) and the audience sizes (A and X) is not less than the secondthreshold, the example audience data calculator 304 determines a uniqueaudience (X) based on a third unique audience sub-process (e.g., hereinreferred to as unique audience sub-process 1.3) (block 514). An examplemanner of implementing the example unique audience sub-process 1.3 isdescribed below in connection with FIG. 5D. The example unique audiencesub-process 1.3 is represented as

$\frac{A^{2}}{R} = {\frac{X^{2}}{T}.}$

Unique audience sub-process 1.3 is based on a mathematical rearrangementof unique audience sub-process 1.2 since the difference between thelogged impressions and the audience size is large

$\left( {{e.g.},{\frac{A^{2}}{R - A} = {{\frac{X^{2}}{T - X}\overset{yields}{\rightarrow}\frac{A^{2}}{R}} = \frac{X^{2}}{T}}}} \right.$

when R»A and T»X). Using unique audience sub-process 1.3, the audiencedata calculator 304 determines the unique audience (X) to be 258

$\left( {{e.g.},{\frac{200^{2}}{600} = \frac{X^{2}}{1000}}} \right).$

At block 516 of FIG. 5A, the example impression data calculator 306(FIG. 3) determines the frequency distribution. The frequencydistribution is the number of people in the unique audience who wereexposed to an impression exactly once, twice, etc. (e.g., the number ofpeople exposed to the media 100 once, the number of people exposed tothe media 100 twice, etc.). The example impression data calculator 306determines the frequency distribution based on a geometric distributionformula as shown below in Equations 10 and 11.

$\begin{matrix}{{{P\left( {Z = k} \right)} = {\left( {1 - p} \right)^{k - 1}p}},} & {{Equation}\mspace{14mu} 10} \\{{p = \frac{X}{T}},} & {{Equation}\mspace{14mu} 11}\end{matrix}$

where k∈{1, 2, . . . , ∞}

In Equations 10 and 11 above, k is the number of impressions per person.Using the above example (e.g., X=270 and T=1000), the example impressiondata calculator 306 determines that 27% of the unique audience (e.g.,27%×270=72.9 people) were exposed to the example media 100 once. Thus,each of those 72.9 users is associated with only one correspondinglogged impression, (e.g., ((1−0.27)¹⁻¹)*0.27=0.27), and 19.7% of theunique audience (e.g., 19%×270=51.3 people) was exposed to the examplemedia 100 twice. Thus, each of those 51.3 users is associated with twocorresponding logged impressions (e.g., ((1−0.27)²⁻¹)*0.27=0.197), etc.

At block 518 of FIG. 5A, the example report generator 308 (FIG. 3)generates a report of the unique audience and/or the frequencydistribution. As described above, the report may include any datarelated to the aggregate database proprietor impression data, thewebsite, demographic data, and/or the logged impressions. In someexamples, the report generator 308 credits the media 100 associated withthe logged impressions based on the unique audience and/or the frequencydistribution.

Turning now to FIG. 5B the example machine readable instructionsrepresented by the flowchart of FIG. 5B may be executed to cause theaudience/impressions determiner 214 of FIG. 3 to determine a totalunique audience (X) using the unique audience sub-process 1.1represented by Equation 12 below.

$\begin{matrix}{\left. {\frac{\left( {R - A} \right)\left( {U - A} \right)}{A^{2}} = \frac{\left( {T - X} \right)\left( {U - X} \right)}{X^{2}}} \right),} & {{Equation}\mspace{14mu} 12}\end{matrix}$

The example instructions represented by the flowchart of FIG. 5B may beused to implement as shown in block 508 of FIG. 5A. The example audiencedata calculator 304 (FIG. 3) determines the total unique audience (X) bysolving the quadratic function corresponding to the example uniqueaudience sub-process 1.1 (e.g., (AR+AU−RU)X²+(A²(−T)−A²U)X+A²TU=0). Inthe illustrated example of the unique audience sub-process 1.1, theexpression AR+AU−RU represents non-mutually exclusive impressions, theexpression A²(−T)−A²U) represents a unique audience sum, and theexpression A²TU represents a unique audience product.

At block 520 of FIG. 5B, the example audience data calculator 304 (FIG.3) multiplies the non-mutually exclusive impressions by the uniqueaudience product to generate a product (e.g., (AR+AU−RU)(A²TU)). Atblock 522, the example audience data calculator 304 determines a squareroot of a difference between a square of the unique audience sum andfour times the product generated at block 520 (e.g., √{square root over((A²(−T)−A²U)²−4(AR+AU−RU)(A²TU))}). At block 524, the example audiencedata calculator 304 sums an opposite of the unique audience sum and thesquare root determined at block 522 (e.g., −(A₂(−T)−A²U)²+√{square rootover ((A²(−T)−A²U)²−4(AR+AU−RU)(A²TU))}). At block 526, the exampleaudience data calculator 304 determines a total unique audience (X)based on dividing the sum at block 524 by two times the non-multipleexclusive impressions, as show in equation 13 below.

$\begin{matrix}{\left. {X = \frac{\begin{matrix}{{- \left( {{A^{2}\left( {- T} \right)} - {A^{2}U}} \right)^{2}} +} \\\sqrt{\left( {{A^{2}\left( {- T} \right)} - {A^{2}U}} \right)^{2} - {4\left( {{AR} + {AU} - {RU}} \right)\left( {A^{2}{TU}} \right)}}\end{matrix}}{2\left( {{AR} + {AU} - {RU}} \right)}} \right),} & {{Equation}\mspace{14mu} 13}\end{matrix}$

Turning now to FIG. 5C, the example machine readable instructionsrepresented by the flowchart of in FIG. 5C may be executed to cause theaudience/impressions determiner 214 of FIG. 3 to determine a totalunique audience (X) using the example unique audience sub-process 1.2represented by Equation 14

$\begin{matrix}{{\frac{A^{2}}{R - A} = \frac{X^{2}}{T - X}},} & {{Equation}\mspace{14mu} 14}\end{matrix}$

The example instructions of FIG. 5B may be used to implement block 512of FIG. 5A. The example audience data calculator 304 (FIG. 3) determinesthe total unique audience (X) by solving the quadratic functioncorresponding to the example unique audience sub-process 1.2 (e.g.,(R−A)X²+A²X−A²T=0). In the illustrated example of the unique audiencesub-process 1.2, the expression R−A represents database proprietorimpression difference, the expression A² represents a databaseproprietor unique audience squared, and the expression A²T represents atotal unique audience product.

At block 528 of FIG. 5C, the example audience data calculator 304 (FIG.3) multiplies the database proprietor impression difference by the totalunique audience product to generate a product (e.g., (R−A)(A²T)). Atblock 530, the example audience data calculator 304 determines a squareroot of a difference between a square of the database proprietor uniqueaudience squared and four times the product generated at block 528(e.g., √{square root over ((A²)²−4(R−A)(A²T))}). At block 532, theexample audience data calculator 304 sums an opposite of the databaseproprietor unique audience squared and the square root determined atblock 530 (e.g., −A²+√{square root over ((A²)²−4(R−A)(A²T))}). At block534, the example audience data calculator 304 determines the totalunique audience (X) based on dividing the sum at determined block 532 bytwo times the database proprietor impression difference, as shown inEquation 15 below,

$\begin{matrix}{{X = \frac{{- A^{2}} + \sqrt{\left( A^{2} \right)^{2} - {4\left( {R - A} \right)\left( {A^{2}T} \right)}}}{2\left( {R - A} \right)}},} & {{Equation}\mspace{14mu} 15}\end{matrix}$

Turning now to FIG. 5D, the example machine readable instructionsrepresented by the flowchart of FIG. 5D may be executed to cause theaudience/impressions determiner 214 of FIG. 3 to determine a totalunique audience (X) by solving the example unique audience sub-process1.3 represented by Equation 16 below.

$\begin{matrix}{{\frac{A^{2}}{R} = \frac{X^{2}}{T}},} & {{Equation}\mspace{14mu} 16}\end{matrix}$

The example instructions of FIG. 5B may be used to implement block 514of FIG. 5A.

At block 536 of FIG. 5D, the example audience data calculator 304 (FIG.3) multiplies the total number of logged impressions (T) by a square ofa total number of registered database proprietor users (A) to generate aproduct (e.g., TA²). In block 538, the example audience data calculator304 divides product by the total number of registered databaseproprietor user impressions (R) to generate a quotient

$\left( {{e.g.},\frac{TA^{2}}{R}} \right).$

At block 540, the audience data calculator 304 determines a total uniqueaudience (X) based on a square root of the quotient. As shown inEquation 17 below.

$\begin{matrix}{{X = \sqrt{\frac{{TA}^{2}}{R}}},} & {{Equation}\mspace{14mu} 17}\end{matrix}$

FIG. 6 is an example flowchart 600 representative of example machinereadable instructions that may be executed to implement the exampleaudience/impressions determiner 214 of FIGS. 2 and 3 to determine aunique audience for the example media 100 associated with two websites(e.g., the example websites 108 of FIG. 1B) and a frequency distributionof the unique audience(s) based on logged impressions of the examplemedia 100 using the example UA/FD process 2 102 (FIG. 1A-1C). Theexample flowchart 600 is described in conjunction with an example shownin FIGS. 8A-8D. FIGS. 8A-8D represent logged impressions associated withthe media 100 accessed via two websites 108 where each user may beexposed to the example media 100 no more than twice per website 108. Asshown in FIGS. 8A-8D, the unique audience for website A is hereinreferred to as “X1,” the unique audience for website B is hereinreferred to as “X2,” and the total unique audience is herein referred toas “X.” Although the example flowchart 600 is described in conjunctionwith an example of a two websites 108 with, at most, two impressions perperson, any number of websites and or logged impressions may be used.

At block 602 of FIG. 6, the example data interface 300 (FIG. 3) obtainsa number of the logged impressions (e.g., database proprietordemographic impressions) and a number of unique users identified by theexample database proprietor 210 (e.g., a partial audience correspondingto registered database proprietor users) in connection with accesses tothe media 100 via a first website (e.g., Website A). In an illustratedexample table 800 of FIG. 8A, the example data interface 300 receives acount of 200 logged impressions that correspond to 150 unique users(e.g., the audience) that accessed the media 100 via website A and thatare registered users of a database proprietor (e.g., the website Apartial audience corresponding to registered database proprietor users).

At block 604 of FIG. 6, the example data interface 300 obtains a numberof logged impressions (e.g., database proprietor demographicimpressions) and a number of unique people (e.g., a partial audience)identified by the example database proprietor 210 as accessing the media100 via a second website (e.g., Website B). As described above inconnection with FIG. 2, the database proprietor 210 that loggedimpressions for website B may or may not be the same as the databaseproprietor that logged impressions for website A. In the illustratedexample table 800 of FIG. 8A, the example data interface 300 obtains acount of 300 logged impressions that correspond to 175 unique peoplethat accessed the media 100 via website B and that are registered usersof the database (e.g., the website B partial audience corresponding toregistered database proprietor users).

At block 606 of FIG. 6, the example data interface 300 obtains a totalnumber of logged impressions (e.g., media impressions) corresponding tothe example client devices 202 and a total number of client device 202in a monitored region (e.g., a universe). In some examples, theimpressions are logged at the server 213 (FIG. 2) based on theimpression requests 206 sent by the client devices 202 (FIG. 2) via thenetwork 204. In some examples, the data interface 300 logs the mediaimpressions corresponding to the media 100 (FIG. 2) accessed at theclient devices 202 (FIG. 2). In some examples, the example datainterface 300 obtains a total number of the logged impressionsassociated with accesses to the media 100 via website A and a totalnumber of logged impressions associated with accesses to the media 100via website B. In the illustrated example table 800 of FIG. 8A, thereare a count of 300 logged impressions associated with website A, a countof 500 logged impressions associated with website B, and a count of 800total logged impressions. Additionally, the universe population in theillustrated example is 1,000 users (not shown).

At block 608 of FIG. 6, the example impression data calculator 306 (FIG.3) generates a constraint matrix (e.g., the example audience/impressionsmodel constraint matrix 819 of FIG. 8C) based on audience and/orimpression constraints and an example constraint vector (e.g., theexample audience/impressions total constraint vector 816 of FIG. 8C)based on the total logged impressions and total audiences in the exampletable 800 of FIG. 8A. In some examples, the constraint matrix 819 isgenerated by gene sub-matrices that represent each constraint value ofthe constraint matrix 819. In the illustrated example of FIGS. 8A-8D,the sub-matrices that represent the constraints are an example totaluniverse audience sub-matrix 804 via any website (FIG. 8B), an exampletotal audience 806 via website A and website B (FIG. 8B), an exampletotal audience for website A 808 (FIG. 8B), an example total audiencefor website B 810 (FIG. 8B), an example total logged impressions forwebsite A 812 (FIG. 8B), and an example total logged impressions forwebsite B 814 (FIG. 8B). Each illustrated example sub-matrix 804, 806,808, 810, 812, 814 includes cells (e.g., c11, c12, c13, etc.) thatrepresent data associated with each constraint. For example, the totaluniverse sub-matrix 804 accounts for all users in a monitored ormeasured region (e.g., the universe) whether or not they were exposed tothe media 100. Therefore, a ‘1’ is placed in every cell to represent anaudience member that was not exposed to the media 100 from eitherwebsite (e.g., c11), an audience member that was only exposed to websiteB once and not to website A (e.g., c12), an audience member that wasexposed to website B twice and not to website A (e.g., c12), etc.

FIG. 8B shows an example audience/impressions constraint matrix 802 thatmay be generated at block 608 of FIG. 6. The audience/impressionsconstraint matrix 802 is a matrix representing various parameters (e.g.,the total population, the total audience for all monitored websites, thetotal audience of website A, etc. to which audience sizes areconstrained. In the example audience/impressions constraint matrix 802,each cell of each constraint sub-matrix 804, 806, 808 810, 812, 814 isrepresented as a row. In some examples, the example impression datacalculator 306 generates the example audience/impressions constraintmatrix 802 at block 608 directly without first generating the examplesub-matrices 804, 806, 808, 810, 812, 814. The example impression datacalculator 306 generates an example audience/impressions total columnvector 816 based on the total number of logged impressions or audienceassociated with constraint in the corresponding row. For example, afirst cell 813 in the example audience/impressions total constraintcolumn vector 814 is the total number of people for the total audience,a second cell 815 is the total audience from website A and B, a thirdcell 817 is the total audience for website A, etc.

At block 610 of FIG. 6, the example impressions data calculator 306applies the audience/impressions constraint matrix 802 and the exampleaudience/impressions total constraint vector 803 to the property ofmaximum entropy distribution to solve the example impressioncharacteristic vector 820 (Q) of Equation 18 below.

maximize Q, H=−Σ _(k=0) ^(∞) q _(k) log(q _(k)),   Equation 18

subject to C_(Q)Q=D_(Q).

Where C_(Q) is the constraint matrix, D_(Q) is the constraint vector,and Q is the impression characteristics. As disclosed above, theimpressions characteristics (Q) include probabilities representing anumber or people associated with zero impressions corresponding towebsite A and/or website B, one impression from website A and/or websiteB, two impressions from website A and/or website B, and/or anycombination thereof. Example phases are shown in FIG. 8C for determininga unique audience and/or a frequency distribution. For example, a firstphase of FIG. 8C at to shows an example equation is populatedillustrating a non-linear equation (e.g., C_(Q)Q=D_(Q)) including anexample constraint matrix (C_(Q)) 819, an example impressioncharacteristic vector (Q) 820, and an example constraint vector (D_(Q))816 after the example impressions data calculator 306 (FIG. 3) appliesthe data of the example table 800 and the example sub-matrices 804-814(FIG. 8B) to the example audience/impression constraint matrix 802 (FIG.8B). The solution for the example impression characteristic vector (Q)820 is shown at time t₁ 822 (FIG. 8C). Each cell in the impressioncharacteristic vector (Q) 820 is associated with a particular cell ofthe example sub-matrices 804-814 (FIG. 8B). For example, impressioncharacteristic q1 corresponds to sub-matrix cell c11 (e.g., the totalnumber of people who were not exposed to the media 100 from eitherwebsite A or website B), the impression characteristic q2 correspondswith sub-matrix cell c21 (e.g., the total number of people that wereexposed to the media 100 accessed via website A once and not exposed tothe media accessed via website B), the impression characteristic q3corresponds with sub-matrix cell c31 (e.g., the total number of peoplethat were exposed to the media 100 accessed via website A twice and notexposed to the media accessed via website B), etc. When the impressiondata calculator 306 multiplies the impression characteristic vector (Q)820 by the universe population (e.g., 1000), the total number of peopleassociated with the particular cell is obtained. For example, 0.2500(e.g., an example impression characteristic q3 821)×1000=25 people whowere exposed to the media 100 twice from website A and 0 times fromwebsite B.

Returning to FIG. 6, at block 612, the example audience data calculator304 (FIG. 3) generates an example population constraint matrix 825 (FIG.8C) based on the sub-matrices associated with the impression constraints(e.g., the example sub-matrices 804, 812, 814 of FIG. 8B) that thepopulation must satisfy. Additionally, the example audience datacalculator 304 generates an example population constraint vector 827(FIG. 8C) based on the total logged impressions associated with thecolumns of the population constraint matrix 825 (FIG. 8C) that thepopulation must satisfy.

At block 614 of FIG. 6, the example audience data calculator 304determines audience characteristics by solving for populationconstraints. The audience characteristics vector (P) 826 includesprobabilities representing the likelihood that a unique audience isassociated with zero impressions corresponding to website A and/orwebsite B, the likelihood that the unique audience is associated withone impression from website A and/or website B, the likelihood that theunique audience is associated with two impressions from website A and/orwebsite B, and/or any combination thereof. For example, the audiencedata calculator 304 applies the population constraint matrix 825 (FIG.8C) and the example population constraint vector 827 (FIG. 8C) to theproperty of maximum entropy distribution to solve the constraints P ofEquation 19 below.

$\begin{matrix}{{{minimize}\mspace{14mu} P},{{D\left( {P\text{:}Q} \right)} = {p_{k}{\log \left( \frac{p_{k}}{q_{k}} \right)}}},{{{subject}\mspace{14mu} {to}\mspace{14mu} C_{P}P} = {D_{P}.}}} & {{Equation}\mspace{14mu} 19}\end{matrix}$

In Equation 19 above, C_(P) is the population constraint matrix based onthe sub-matrices associated with the impression constraints (e.g., theexample sub-matrices 804, 812, 814 of FIG. 8B) that the population mustsatisfy, D_(P) is the population constraint vector, and populationconstraints (P) representative of the probabilities associated with theunique audience for website A, website B, and the total unique audience.Example phases are shown in FIG. 8C for determine audiencecharacteristics. The second phase at time t₂ 824 is an example phasethat follows the first phase at time t₁ 822. For example, the secondphase of FIG. 8C at time t₂ 824 shows an example manner in which theaudience data calculator 304 can generate an example equationillustrating the non-linear equation (e.g., C_(P)P=D_(P)) including theexample population constraint matrix (C_(P)) 825, an example audiencecharacteristic vector (D_(P)) 827, and the example audience constraintvector (P) 826. The solution for the example audience characteristicvector 826 is shown at time t₃ 828 of FIG. 8C.

At block 616 of FIG. 6, the example audience data calculator 304 appliesthe audience characteristics to relevant constraints. For example, therelevant constraints are constraints related to the impressionscorresponding to an audience of website A and website B (e.g., the totalaudience sub-matrix 806, the example impressions for website Asub-matrix 812, and the example sub-matrix for website B 814). Examplephases are shown in FIG. 8D to determine a unique audience for websiteA, a unique audience for website B, and a total unique audience. Thefourth phase at time t₄ 830 is an example phase that follows the thirdphase at time t₃ 828. For example, the fourth phase of FIG. 8D at timet₄ 830 shows an example equation representing how the unique audience isgenerated, the example unique audience matrix 831 has three rows torepresents the three unique audience constraints (e.g., the exampletotal audience 806, the example total audience for site A 808, and theexample total unique audience for site B 810). The product of theaudience constraints 831 and the example audience characteristics 826 isan example unique audience vector 832 representative of theprobabilities associated with the unique audience for website A, websiteB, and the total unique audience. In such an example, the uniqueaudience vector 832 may be multiplied by the universe population (e.g.,1000) to determine the unique audience for website A (e.g., 218), theunique audience for website B (e.g., 281), and the total unique audience(e.g., 363) for the example media 100, as shown at time t₅ 834.

At block 618 of FIG. 6, the example report generator 308 (FIG. 3)generates a report for the unique audience and/or the frequencydistribution. As described above, the report may include any datarelated to the aggregate database proprietor impression data, themonitored website(s), demographic data, and/or the logged mediaimpressions. In some examples, the report generator 308 credits mediaassociated with impression data in the report based on the uniqueaudience and/or the frequency distribution. For example, the reportgenerator 308 may credit the media 100 (FIGS. 1A-1C and 2) by storingone or more of any type of impression data and/or demographic impressiondata in association with a media identifier of the media 100 in amachine readable memory (e.g., one or more of the memories 1014, 1016 ofFIG. 10).

FIG. 7 is an example flowchart 700 representative of example machinereadable instructions that may be executed to implement the exampleaudience/impressions determiner 214 of FIGS. 2 and 3 to determine aunique audience for the example media 100 associated with two websites(e.g., the example websites 109 of FIG. 1C) and/or a frequencydistribution of the unique audience(s) based on logged impressions ofthe media 100 using the example UA/FD process 3 103 (FIG. 1A-1C). Theexample flowchart is described in conjunction with an example shown inFIG. 9. FIG. 9 represents logged impressions associated with the media100 accessed via a large number of websites (e.g., the example websites109) where each user may be exposed to the media 100 any number oftimes. As shown in FIG. 9, the unique audience for website A is hereinreferred to as “X1,” the unique audience for website B is hereinreferred to as “X2,” and the total unique audience is herein referred toas “X.” Although the example flowchart 700 is described in conjunctionwith an example of two websites with an infinite number of loggedimpressions website, any number of websites and/or logged impressionsmay be used.

At block 702 of FIG. 7, the example data interface 300 of FIG. 3 obtainsa number of logged media impressions (e.g., database proprietordemographic impressions) and a number of unique people identified by theexample database proprietor 210 in connection with accesses to theexample media 100 via a first website (e.g., website A). In theillustrated example table 900 of FIG. 9, the example receiver obtains acount of 200 logged database proprietor demographic impressions thatcorrespond to 150 unique people (e.g., the audience) that accessed themedia 100 via website A and that are registered users of the databaseproprietor (e.g., the website A partial audience corresponding toregistered database proprietor users) associated with the media 100.

At block 704 of FIG. 7, the example data interface 300 obtains a numberof logged impression (e.g., database proprietor demographic impressions)and a number of unique people identified by the example databaseproprietor 210 as accessing the example media 100 via a second website(e.g., Website B). As described above in connection with FIG. 2, thedatabase proprietor 210 that logged impressions for website B may or maynot be the same as the database proprietor 210 that logged impressionsfor website A. In the illustrated example table 900 of FIG. 9, theexample receiver obtains a count of 300 database proprietor demographiclogged impressions that correspond to 175 unique people that accessedthe media 100 via website B and that are registered users of thedatabase (e.g., the website B partial audience corresponding toregistered database proprietor users).

At block 706 of FIG. 7, the example data interface 300 obtains a totalnumber of impressions (e.g., media impressions) corresponding to theexample client devices 202 and a total number of client devices 202 in amonitored region (e.g., a universe). In some examples, the impressionsare logged by server 213 (FIG. 2) based on the impression requests 206sent by client devices 202 (FIG. 2) via the network 204. In someexamples, the data interface 300 logs the media impressionscorresponding to the media 100 (FIG. 2) accessed at the client devices202 (FIG. 2). In some examples, the example data interface 300 receivesa total number or count of the logged impressions associated withaccesses to the media 100 via website A and a total number or count ofthe logged impressions associated with accesses to the media 100 viawebsite B. Example phases are shown in FIG. 9 for determining a uniqueaudience and/or frequency distribution for media impressions. Forexample, a first phase of FIG. 9 at time t₀ 900 shows an example tablethat includes a count of 300 logged database proprietor demographicimpressions associated with website A, a count of 500 logged databaseproprietor demographic impressions associated with website B, and acount of 800 total logged media impressions. Additionally, the universepopulation in the illustrated example is 1,000 users (not shown).

At block 708 of FIG. 7, the example impression data determiner 306 (FIG.3) calculates an enumeration table based on the data in the exampletable 900 (FIG. 9) and the principles of an infinite geometric series.Based on derivations of the infinite geometric series, a value for eachentry in an example combination table can be determined shown at time t₁902 (FIG. 9). Each entry is determined based on the following formulafor logged impressions:

${\sum\limits_{i = 1}^{\infty}{\sum\limits_{j = 1}^{\infty}{\sum\limits_{k = 1}^{\infty}{z_{1}^{({a_{0} + {a_{1}i} + {a_{2}j} + {a_{3}k}})}z_{2}^{({b_{0} + {b_{1}i} + {b_{2}j} + {b_{3}k}})}}}}} = \frac{z_{1}^{({a_{0} + a_{1} + a_{2} + a_{3}})}z_{2}^{({b_{0} + b_{1} + b_{2} + b_{3}})}}{\left( {1 - {z_{1}^{a_{1}}z_{2}^{b_{1}}}} \right)\left( {1 - {z_{1}^{a_{2}}z_{2}^{b_{2}}}} \right)^{2}\left( {1 - {z_{1}^{a_{3}}z_{2}^{b_{3}}}} \right)}$

Where variable z_(i) is representative of the i^(th) website.

The following formula is used for the audiences:

${\sum\limits_{i = 1}^{\infty}{\sum\limits_{j = 1}^{\infty}{\sum\limits_{k = 1}^{\infty}{z_{1}^{({a_{0} + {a_{1}i} + {a_{2}j} + {a_{3}k}})}z_{2}^{({b_{0} + {b_{1}i} + {b_{2}j} + {b_{3}k}})}}}}} = {\frac{z_{1}^{({a_{0} + a_{1} + a_{2} + a_{3}})}z_{2}^{({b_{0} + b_{1} + b_{2} + b_{3}})}}{\left( {1 - {z_{1}^{a_{1}}z_{2}^{b_{1}}}} \right)\left( {1 - {z_{1}^{a_{2}}z_{2}^{b_{2}}}} \right)^{2}\left( {1 - {z_{1}^{a_{3}}z_{2}^{b_{3}}}} \right)}.}$

For example, the value associated with impression combination I2 in theexample combination table is represented by:

${\sum\limits_{i = 1}^{\infty}{i \times \left( z_{1}^{(R)} \right)^{i}z_{1}^{(A)}z_{A}z_{*}}} = \frac{z_{1}^{(R)}z_{1}^{(A)}z_{A}z_{*}}{\left( {1 - z_{1}^{(R)}} \right)^{2}}$

Additionally, the value associated with impression combination I3 in theexample combination table is represented by:

${\sum\limits_{j = 1}^{\infty}{\sum\limits_{i = 1}^{\infty}{i*\left( z_{i}^{(R)} \right)^{i}\left( z_{2}^{(R)} \right)^{j}z_{1}^{(A)}z_{2}^{(A)}z_{*}}}} = \frac{z_{1}^{(R)}z_{2}^{(R)}z_{1}^{(A)}z_{2}^{(A)}z_{A}z_{*}}{\left( {1 - z_{1}^{(R)}} \right)^{2}\left( {1 - z_{2}^{(R)}} \right)}$

Since a sum of the impression combinations I2, I3 (e.g., I2+I3)represents all the logged database proprietor demographic impressionsaccessed via website A, then I2+I3=200/1000 (e.g., the logged databaseproprietor demographic impressions accessed via website A/the universalpopulation). Additional equations can be determined using a similarmanner (e.g., I1+I4=300/1000, etc.) The result is N equations (e.g., onefor each constraint) with N unknowns (e.g. the z variables). A thirdphase of FIG. 9 at time t₂ 904 shows an example manner in which theexample impressions data calculator 306 (FIG. 3) may determine anenumeration table based on a solution of the non-linear system ofequation. The enumeration table completely describes the loggedimpressions and audience recorded by the database proprietor. Theenumeration table is representative of logged database proprietordemographic impressions across the registered users of each website.

A fourth phase of FIG. 9 at time t₃ 906 show an example manner in whichthe example audience data determiner 304 may calculate an examplepopulation enumeration table (block 712). The example populationenumeration table represents the estimated distribution of loggedimpression across the total unique audience. The example populationenumeration table may be used to estimate the population parameterswhere some of the z variables will be related to known populationconstraints (block 712). The population constraints are representativeof a total of logged impressions for media accessed via each website(e.g., both impressions logged by the database proprietor and not loggedby the database proprietor). For example, as shown in the table at timet₀, website A includes a total of 200 database proprietor demographicimpressions recorded (e.g., logged) by a database proprietor and a totalof 300 impressions for the population (e.g., 100 impressions were notlogged by the database proprietor). Based on derivations of the infinitegeometric series, the example impression data calculator 306 (FIG. 3)determines a value for each entry in an example combination table shownat time t₁ 902 (FIG. 9). For example, the value associated with audiencecombination A2 in the example combination table shown at time t₁ 902(FIG. 9) is represented by:

${\sum\limits_{i = 1}^{\infty}{i \times \left( z_{1}^{(T)} \right)^{i}z_{1}^{(X)}z_{X}z_{*}}} = \frac{z_{1}^{(T)}z_{2}^{(A)}z_{X}{\overset{\bigvee}{z}}_{*}}{\left( {1 - z_{1}^{(T)}} \right)^{2}}$

Additionally, the value associated with audience combination A3 in theexample combination table shown at time t₂ 904 is represented by:

${\sum\limits_{j = 1}^{\infty}{\sum\limits_{i = 1}^{\infty}{i*\left( z_{1}^{(T)} \right)^{i}\left( z_{2}^{(T)} \right)^{j}z_{1}^{(X)}z_{2}^{(X)}z_{*}}}} = \frac{z_{1}^{(T)}z_{2}^{(T)}z_{1}^{(T)}z_{2}^{(T)}z_{X}{\overset{\bigvee}{z}}_{*}}{\left( {1 - z_{1}^{(T)}} \right)^{2}\left( {1 - z_{2}^{(T)}} \right)}$

Since the sum of the audience combinations A2, A3 (e.g., A2+A3)represents the audience exposed to the media 100 accessed via website A,then A2+A3=300/1000 (e.g., the logged impressions associated with thepopulation audience of website A/the universal population). Additionalequations can be determined using a similar manner (e.g.,A1+A4=500/1000, etc.) This particular mathematical problem creates threeequations with three unknowns (e.g., z₁ ^((X)), z₂ ^((X)), ž.). Theexample audience data calculator 304 (FIG. 3) solves for any unknownsfor each combination based on the constraints (e.g., A2+A3=300/1000,A1+A4=500/1000, etc.). For example, the example audience data calculator304 may determine the example population enumeration table of time t₃906 of FIG. 9 by solving the non-linear system of equation for the threeunknowns (block 714). The example population enumeration tablecompletely describes the logged impressions and audience recorded by thedatabase proprietor. Any value of interest from that distribution (e.g.,frequency distribution, audiences, or conditional probabilities, can bedetermined using the z values of the example enumeration table (e.g.,optimizing both memory and speed).

At block 716 of FIG. 7, the example audience data determiner 304 (FIG.3) applies the unknowns to audience data (e.g., the unique audience forwebsite A, the unique audience for website B, etc.). For example, theexpressions of the cells of the example table shown at time t₁ 902 (FIG.9) associated with the example audience (e.g., A1, A2, A3, A4, and zA)are shown in an example population table shown (e.g., i, ii, iii, iv, v,vi, and vii) at time t₄ 908 (FIG. 9). After a value for each cell hasbeen calculated, the appropriate cells are summed and multiplied by thetotal universe (UA) to determine the unique audience from website A, theunique audience of website B, and the total unique audience, as shownbelow and in an example result table shown at time t₅ 910 (FIG. 9):

X₁=UA ((i)+(ii)=1000 (0.08654+0.122699)=209.24 (e.g., unique audience ofwebsite A)

X₂=UA ((iii)+(iv)=1000 (0.125720+0.122699)=248.42 (e.g., unique audienceof website B)

X₃=UA ((v)+(vi)+(vii))=1000 (0.08654+0.125720+0.122699)=334.96 (e.g.,total unique audience)

At block 718 of FIG. 7, the example report generator 308 (FIG. 3)generates a report that includes the unique audience and/or thefrequency distribution. As described above, the report may include anydata related to the aggregate database proprietor impression data, thewebsite, demographic data, and/or the logged impressions. In someexamples, the report generator 308 (FIG. 3) credits media associatedwith a logged impression based on the unique audience and/or thefrequency distribution. For example, the report generator 308 may creditthe media 100 (FIGS. 1A-1C and 2) by storing one or more of any type ofimpression data and/or demographic impressions data in association witha media identifier of the media 100 in a machine readable memory (e.g.,one or more of the memories 1014, 1016 of FIG. 10).

FIG. 10 is a block diagram of an example processor platform 1000 capableof executing the instructions of FIGS. 4-7 to implement the examplememory controller 202 of FIG. 2. The processor platform 1000 can be, forexample, a server, a personal computer, a mobile device (e.g., a cellphone, a smart phone, a tablet such as an iPad™), a personal digitalassistant (PDA), an Internet appliance, or any other type of computingdevice.

The processor platform 1000 of the illustrated example includes aprocessor 1012. The processor 1012 of the illustrated example ishardware. For example, the processor 1012 can be implemented byintegrated circuits, logic circuits, microprocessors or controllers fromany desired family or manufacturer.

The processor 1012 of the illustrated example includes the examplememory 212 (e.g., a cache). The example processor 1012 of FIG. 10executes the instructions of FIGS. 4-7 to implement the example datainterface 300, the example process selector 302, the example audiencedata calculator 304, the example impression data calculator 306, and theexample report generator 308 of FIG. 3 to implement the exampleaudience/impression determiner 214. The processor 1012 of theillustrated example is in communication with a main memory including avolatile memory 1014 and a non-volatile memory 1016 via a bus 1018. Thevolatile memory 1014 may be implemented by Synchronous Dynamic RandomAccess Memory (SDRAM), Dynamic Random Access Memory (DRAM), RAMBUSDynamic Random Access Memory (RDRAM) and/or any other type of randomaccess memory device. The non-volatile memory 1016 may be implemented byflash memory and/or any other desired type of memory device. Access tothe main memory 1014, 1016 is controlled by a memory controller.

The processor platform 1000 of the illustrated example also includes aninterface circuit 1020. The interface circuit 1020 may be implemented byany type of interface standard, such as an Ethernet interface, auniversal serial bus (USB), and/or a PCI express interface.

In the illustrated example, one or more input devices 1022 are connectedto the interface circuit 1020. The input device(s) 1022 permit(s) a userto enter data and commands into the processor 1012. The input device(s)can be implemented by, for example, a sensor, a microphone, a camera(still or video), a keyboard, a button, a mouse, a touchscreen, atrack-pad, a trackball, isopoint and/or a voice recognition system.

One or more output devices 1024 are also connected to the interfacecircuit 1020 of the illustrated example. The output devices 1024 can beimplemented, for example, by display devices (e.g., a light emittingdiode (LED), an organic light emitting diode (OLED), a liquid crystaldisplay, a cathode ray tube display (CRT), a touchscreen, a tactileoutput device, and/or speakers). The interface circuit 1020 of theillustrated example, thus, typically includes a graphics driver card, agraphics driver chip or a graphics driver processor.

The interface circuit 1020 of the illustrated example also includes acommunication device such as a transmitter, a receiver, a transceiver, amodem and/or network interface card to facilitate exchange of data withexternal machines (e.g., computing devices of any kind) via a network1026 (e.g., an Ethernet connection, a digital subscriber line (DSL), atelephone line, coaxial cable, a cellular telephone system, etc.).

The processor platform 1000 of the illustrated example also includes oneor more mass storage devices 1028 for storing software and/or data.Examples of such mass storage devices 1028 include floppy disk drives,hard drive disks, compact disk drives, Blu-ray disk drives, RAIDsystems, and digital versatile disk (DVD) drives.

The coded instructions 1032 of FIGS. 4-7 may be stored in the massstorage device 1028, in the volatile memory 1014, in the non-volatilememory 1016, and/or on a removable tangible computer readable storagemedium such as a CD or DVD.

From the foregoing, it will be appreciated that the above disclosedexamples may be used to select a UA/FD process that based on bothprocessor resources and memory resources to provide estimations ofunique audiences (e.g., a unique audience for a particular websiteand/or a total unique audience) that is more accurate than priortechniques by using counts of logged impressions and a number of uniquepeople identified by a database proprietor (e.g., a partial audience).Using examples disclosed herein, unique audiences can be determined in afaster and more accurate manner with less required memory resources.

Prior techniques for determining a unique audience for a mediapresentation include simple scaling or negative binomial distribution.However, simple scaling is inaccurate, negative binomial distributionincludes computations requiring a large amount of processor resources,and both simple scaling and negative binomial distribution cannotdetermine a unique audiences for particular websites and a total uniqueaudience. Examples disclosed herein alleviate such problems by selectingone of three UA/FD processes based on available processor resources andmemory resources to determine a unique audience and/or a frequencydistribution of impressions for a media presentation. Additionally, twoof the three UA/FD processes disclosed herein determine a uniqueaudience based on instructions to measure media exposure accessed viamore than one website leading to more accurate estimations.

Example methods are disclosed to determine a unique audience exposed tomedia. Such example methods include receiving impression requests at aserver from a plurality of client devices via a network. Such examplemethods include, based on the impression requests, logging, with theserver, a plurality of impressions corresponding to the media accessedat the client devices; obtaining a number of demographic impressionslogged by a database proprietor; obtaining a number of registered usersof the database proprietor exposed to the media; multiplying, byexecuting an instruction with a processor, a number of the plurality ofimpressions by a square of the number of the registered users togenerate a product; dividing, by executing an instruction with theprocessor, the product by the number of the demographic impressions togenerate a quotient; and determining, by executing an instruction withthe processor, the unique audience based on a square root of thequotient.

In some example methods, the registered users of the database proprietorexposed to the media correspond to the demographic impressions and to atleast some of the plurality of impressions. In some examples, thecrediting of the media is associated with the plurality of impressionsbased on the unique audience. In some examples, a frequency distributionis determined for the plurality of impressions based on the uniqueaudience by: dividing the unique audience by the number of the pluralityof impressions to determine a second quotient; and calculating ageometric distribution based on the second quotient.

Example methods are disclosed to determine a unique audience exposed tomedia. Such example methods include receiving impression requests at aserver from a plurality of client devices via a network. Such examplemethods include, based on the impression requests, logging, with theserver, a plurality of impressions corresponding to the media accessedat the client devices; obtaining a first number of first demographicimpressions corresponding to the media accessed via a first website andlogged by a first database proprietor, the first demographic impressionscorresponding to first registered users of the first databaseproprietor; obtaining a second number of second demographic impressionscorresponding to the media accessed via a second website and identifiedby a second database proprietor, the second demographic impressionscorresponding to second registered users of the second databaseproprietor; obtaining a first number of the first registered usersexposed to the media; obtaining a second number of the second registeredusers exposed to the media; generating, by executing an instruction witha processor, a constraint matrix and a constraint vector based on thefirst number of the first demographic impressions and the second numberof the second demographic impressions, the constraint vectorrepresentative of a plurality of ratios of constraints to a number ofthe plurality of impressions; determining, by executing an instructionwith the processor, audience characteristics based on the constraintmatrix and the constraint vector; and determining, by executing aninstruction with the processor, a first unique audience exposed to themedia via the first website, a second unique audience exposed to themedia via the second website, and a total unique audience exposed to themedia via the first and second websites based on the audiencecharacteristics.

In some example methods, the total unique audience is a count of uniqueaudience members across the first unique audience and the second uniqueaudience. In some example methods, the constraint matrix is a firstconstraint matrix and the constraint vector is a first constraintvector, and further including determining the first unique audience, thesecond unique audience, and the total unique audience by: generating thefirst constraint matrix and the first constraint vector based on thefirst number of the first demographic impressions, the first number ofthe first registered users, the second number of the second demographicimpressions, and the second number of the second registered users;determining the impression characteristics associated with the firstconstraint matrix and the first constraint vector, the impressioncharacteristics including maximized values based on the first constraintmatrix and the first constraint vector; and generating a secondconstraint matrix and a second constraint vector based on the impressioncharacteristics, the audience characteristics including maximized valuesbased on the second constraint matrix and the second constraint vector,the first unique audience, the second unique audience, and the totalunique audience being based on the audience characteristics.

In some example methods, a frequency distribution is determined based onthe impression characteristics. In some example methods, the impressioncharacteristics include probabilities representing likelihoods ofdifferent numbers of people exposed to the media via at least one of thefirst website or the second website. In some example methods, theaudience characteristics include probabilities representing likelihoodsof different sizes of unique audiences corresponding to at least one ofthe first website or the second website. In some example methods, thefirst database proprietor is the second database proprietor. In someexample methods, a report is generated indicating at least one of thefirst unique audience, the second unique audience, or the total uniqueaudience.

In some example methods, the media associated with the plurality ofimpressions is credited based on at least one of the first uniqueaudience, the second unique audience, or the total unique audience. Insome example methods, the constraint matrix includes constraintsrepresented in sub-matrices, the constraints including at least one of afirst size of a first audience exposed to the media via the firstwebsite, a second size of a second audience exposed to the media via thesecond website, a third size of a total audience exposed to the mediavia the first and second websites, a fourth size of a universe audiencevia any website, a first count of first impressions corresponding to themedia accessed via the first website, and a second count of secondimpressions corresponding to the media accessed via the second website.

Example methods are disclosed to determine a unique audience exposed tomedia. Such example methods include receiving impression requests at aserver from a plurality of client devices via a network. Such examplesinclude, based on the impression requests, logging, with the server, aplurality of impressions corresponding to the media accessed at theclient devices; obtaining a first number of first demographicimpressions corresponding to the media accessed via a first website andlogged by a first database proprietor, the first demographic impressionscorresponding to first registered users of the first databaseproprietor; obtaining a second number of second demographic impressionscorresponding to the media accessed via a second website and logged by asecond database proprietor, the second demographic impressionscorresponding to second registered users of the second databaseproprietor; obtaining a first number of the first registered usersexposed to the media; obtaining a second number of the second registeredusers exposed to the media; determining, by executing an instructionwith a processor, a first enumeration table, the first enumeration tableincluding first values based on a system of non-linear equationsassociated with the first number of the first demographic impressions,the first number of the first registered users, the second number of thesecond demographic impressions and the second number of the secondregistered users; determining, by executing an instruction with theprocessor, a second enumeration table including second values based on asecond system of non-linear equations associated with the first valuescalculated in the first enumeration table and the plurality ofimpressions; and determining, by executing an instruction with theprocessor, a first unique audience of the media accessed via the firstwebsite, a second unique audience of the media accessed via the secondwebsite, and a total unique audience of the media accessed via the firstand second websites using second expressions solved based on the secondvalues in the second enumeration table.

In some example methods, the first enumeration table is an estimateddistribution of the plurality of impressions across the first registeredusers and the second registered users. In some example methods, thesecond enumeration table is an estimated distribution of the pluralityof impressions across the total unique audience. In some examplemethods, a frequency distribution of impressions for the total uniqueaudience is determined based on the second enumeration table.

In some example methods, the second enumeration table is based onpopulation constraints. In such example methods, the populationconstraints are representative of: a third number of the firstdemographic impressions corresponding to the media accessed via thefirst website, the third number of the first demographic impressionsincluding the first number of the first demographic impressions; and afourth number of the second demographic impressions corresponding to themedia accessed via the second website, the fourth number of the seconddemographic impressions including the second number of the seconddemographic impressions.

In some examples, the first database proprietor is the second databaseproprietor. In some examples, a report is generated indicating at leastone of the first unique audience, the second unique audience, or thetotal unique audience. In some examples, the media associated with theplurality of impressions is credited based on at least one of the firstunique audience, the second unique audience, or the total uniqueaudience.

Although certain example methods, apparatus and articles of manufacturehave been described herein, the scope of coverage of this patent is notlimited thereto On the contrary, this patent covers all methods,apparatus and articles of manufacture fairly falling within the scope ofthe claims of this patent.

1. (canceled)
 2. An apparatus comprising: a server to, based onimpression requests from a plurality of client devices via a network,log a plurality of impressions corresponding to media accessed at theclient devices; an interface to: obtain a count of demographicimpressions logged by a database proprietor; and obtain a count ofregistered users of the database proprietor exposed to the media; and aprocessor to execute instructions to determine a unique audience sizeby: multiplying a count of the plurality of impressions by a square ofthe count of the registered users to generate a product; dividing theproduct by the count of the demographic impressions to generate aquotient; and determining the unique audience size based on a squareroot of the quotient.
 3. The apparatus of claim 2, wherein the processoris to determine the unique audience size based on the square root of thequotient when the unique audience size is not available from a server ofthe database proprietor.
 4. The apparatus of claim 2, wherein theinstructions are first instructions, the processor to select the firstinstructions to determine the unique audience size instead of secondinstructions to determine the unique audience size, the firstinstructions to cause more resource-efficient operation of a computerthan the second instructions by utilizing less processor resources andmemory resources than the second instructions.
 5. The apparatus of claim2, wherein the registered users of the database proprietor exposed tothe media correspond to the demographic impressions and to at least someof the plurality of impressions.
 6. The apparatus of claim 2, whereinthe processor is to generate a report indicating the unique audiencesize.
 7. The apparatus of claim 2, wherein the processor is to creditthe media associated with the plurality of impressions based on theunique audience size.
 8. The apparatus of claim 2, wherein the processoris to determine a frequency distribution for the plurality ofimpressions based on the unique audience size by: dividing the uniqueaudience size by the count of the plurality of impressions to determinea second quotient; and calculating a geometric distribution based on thesecond quotient.
 9. An apparatus comprising: a server to, based onimpression requests from a plurality of client devices via a network,log a plurality of impressions corresponding to media accessed at theclient devices; an interface to: obtain a first count of firstdemographic impressions corresponding to the media accessed via a firstwebsite and logged by a first database proprietor, the first demographicimpressions corresponding to first registered users of the firstdatabase proprietor; obtain a second count of second demographicimpressions corresponding to the media accessed via a second website andidentified by a second database proprietor, the second demographicimpressions corresponding to second registered users of the seconddatabase proprietor; obtain a first count of the first registered usersexposed to the media; and obtain a second count of the second registeredusers exposed to the media; and a processor to: generate a constraintmatrix and a constraint vector based on the first count of the firstdemographic impressions and the second count of the second demographicimpressions, the constraint vector representative of a plurality ofratios of constraints to a count of the plurality of impressions; anddetermine a first unique audience size exposed to the media via thefirst website, a second unique audience size exposed to the media viathe second website, and a total unique audience size exposed to themedia via the first and second websites based on the constraint matrixand the constraint vector.
 10. The apparatus of claim 9, wherein theprocessor is to determine the first unique audience size, the secondunique audience size, and the total unique audience size when the firstunique audience size, the second unique audience size, and the totalunique audience size are not available from a server of the databaseproprietor.
 11. The apparatus of claim 9, wherein the total uniqueaudience size is a count of unique audience members across the firstunique audience size and the second unique audience size.
 12. Theapparatus of claim 9, wherein the constraint matrix is a firstconstraint matrix and the constraint vector is a first constraintvector, and the processor is to determine the first unique audiencesize, the second unique audience size, and the total unique audiencesize by: generating the first constraint matrix and the first constraintvector based on the first count of the first demographic impressions,the first count of the first registered users, the second count of thesecond demographic impressions, and the second count of the secondregistered users; determining impression characteristics associated withthe first constraint matrix and the first constraint vector, theimpression characteristics including maximized values based on the firstconstraint matrix and the first constraint vector; and generating asecond constraint matrix and a second constraint vector based on theimpression characteristics, the first and second constraint matrices andthe first and second constraint vectors corresponding to audiencecharacteristics, the audience characteristics including maximized valuesbased on the second constraint matrix and the second constraint vector,the first unique audience size, the second unique audience size, and thetotal unique audience size being based on the audience characteristics.13. The apparatus of claim 12, wherein the processor is to determine afrequency distribution based on the impression characteristics.
 14. Theapparatus of claim 12, wherein the impression characteristics includeprobabilities representing likelihoods of different numbers of peopleexposed to the media via at least one of the first website or the secondwebsite.
 15. The apparatus of claim 12, wherein the audiencecharacteristics include probabilities representing likelihoods ofdifferent sizes of unique audiences corresponding to at least one of thefirst website or the second website.
 16. The apparatus of claim 9,wherein the first database proprietor is the second database proprietor.17. The apparatus of claim 9, wherein the processor is to generate areport indicating at least one of the first unique audience size, thesecond unique audience size, or the total unique audience size.
 18. Theapparatus of claim 9, wherein the processor is to credit the mediaassociated with the plurality of impressions based on at least one ofthe first unique audience size, the second unique audience size, or thetotal unique audience size.
 19. The apparatus of claim 9, wherein theconstraint matrix includes constraints represented in sub-matrices, theconstraints including at least one of a first size of a first audienceexposed to the media via the first website, a second size of a secondaudience exposed to the media via the second website, a third size of atotal audience exposed to the media via the first and second websites, afourth size of a universe audience via any website, a first count offirst impressions corresponding to the media accessed via the firstwebsite, and a second count of second impressions corresponding to themedia accessed via the second website.
 20. An apparatus comprising: aserver to, based on impression requests from a plurality of clientdevices via a network, log a plurality of impressions corresponding tomedia accessed at the client devices; an interface to: obtain a firstcount of first demographic impressions corresponding to the mediaaccessed via a first website and logged by a first database proprietor,the first demographic impressions corresponding to first registeredusers of the first database proprietor; obtain a second count of seconddemographic impressions corresponding to the media accessed via a secondwebsite and logged by a second database proprietor, the seconddemographic impressions corresponding to second registered users of thesecond database proprietor; obtain a first count of the first registeredusers exposed to the media; and obtain a second count of the secondregistered users exposed to the media; and a processor to: determine afirst enumeration table, the first enumeration table including firstvalues based on a system of non-linear equations associated with thefirst count of the first demographic impressions, the first count of thefirst registered users, the second count of the second demographicimpressions and the second count of the second registered users;determine a second enumeration table including second values based on asecond system of non-linear equations associated with the first valuescalculated in the first enumeration table and the plurality ofimpressions; and determine a first unique audience size of the mediaaccessed via the first website, a second unique audience size of themedia accessed via the second website, and a total unique audience sizeof the media accessed via the first and second websites using secondexpressions solved based on the second values in the second enumerationtable.
 21. The apparatus of claim 20, wherein the processor is todetermine the first unique audience size, the second unique audiencesize, and the total unique audience size when the first unique audiencesize, the second unique audience size, and the total unique audiencesize are not available from a server of the database proprietor.
 22. Theapparatus of claim 20, wherein the first enumeration table is anestimated distribution of the plurality of impressions across the firstregistered users and the second registered users.
 23. The apparatus ofclaim 20, wherein the second enumeration table is an estimateddistribution of the plurality of impressions across the total uniqueaudience size.
 24. The apparatus of claim 23, wherein the processor isto determine a frequency distribution of impressions for the totalunique audience size based on the second enumeration table.
 25. Theapparatus of claim 20, wherein the second enumeration table is based onpopulation constraints, the population constraints representative of: athird count of the first demographic impressions corresponding to themedia accessed via the first website, the third count of the firstdemographic impressions including the first count of the firstdemographic impressions; and a fourth count of the second demographicimpressions corresponding to the media accessed via the second website,the fourth count of the second demographic impressions including thesecond count of the second demographic impressions.
 26. The apparatus ofclaim 20, wherein the first database proprietor is the second databaseproprietor.
 27. The apparatus of claim 20, wherein the processor is togenerate a report indicating at least one of the first unique audiencesize, the second unique audience size, or the total unique audiencesize.
 28. The apparatus of claim 20, wherein the processor is to creditthe media associated with the plurality of impressions based on at leastone of the first unique audience size, the second unique audience size,or the total unique audience size.